Chapter I
Understanding Spatial and Size Relationships
Why is it worth developing spatial and size relationships?
Understanding spatial and size relationships is one of the key mathematical competencies developed in the early school years. For preschool and early-school children, stating the position of an object precisely is a real challenge, because laterality is still developing. Supporting this process is especially important at this stage, as it affects not only mathematical competence but also the acquisition of reading and writing. A child who can name directions, the position of objects in space, and their size gains foundations not only for learning geometry, but also for solving everyday problems — from moving around in the environment, through organizing space, to planning actions in time and space.
These skills are closely linked to the development of logical thinking, spatial orientation, and geometric imagination. Building them also supports language development — through using terms such as: above, below, next to, between, farther, higher, lower, longer, shorter, and so on. A correct understanding of these ideas is the basis for learning more advanced mathematical and technical topics later on.
- shaping logical thinking
- developing spatial orientation
- developing geometric imagination
- enriching speech and vocabulary
Working with blocks in this area proves especially effective. Thanks to physical, three-dimensional pieces — such as platforms, cylinders, and gears — children can independently construct, test, compare, and modify their buildings. Through action, handling objects, and observing the effects of their position and relations to one another, pupils intuitively develop an understanding of space and measure. Blocks make it possible not only to build and experiment, but also to observe the link between movement (for example, the turning of gears) and changes in a spatial arrangement.
- differentiating the level of difficulty
- adapting to individual pupils' needs
- introducing elements of coding and planning actions
The construction activities proposed in this chapter integrate movement, thinking, imagination, and cooperation. We will practice telling right from left. We will order constructions by size. We will develop fine motor skills, and through group work we will practice communication skills.
After completing the tasks in this chapter, the pupil:
- Describes the position of pieces in space, using the terms: above, below, next to, between, in front of, behind, on the left, on the right.
- Distinguishes and uses directional and locational terms (e.g. to the right, to the left, up, down), building constructions from spoken or picture instructions.
- Compares the length, height, and width of pieces and of constructions built from blocks.
- Estimates the size of objects and the distances between them, using the terms: shorter, longer, the same, taller, lower, wider, narrower.
- Builds spatial constructions from the teacher's instructions, a plan, or their own design.
- Reproduces and creates spatial arrangements using different types of pieces: gears, cylinders, platforms.
- Analyzes and describes spatial relations between pieces of a construction — e.g. what is between two gears, what is higher or farther away.
- Designs spatially complex routes, mazes, and construction layouts, planning the direction and sequence of moves.
- Makes simple measurements (e.g. the length of a construction) and records or compares the results.
- Works in a team, communicating precisely while building shared models.
Top – Bottom – Middle
Learning goal
Recognizing the position of objects in space: on, under, next to, in the middle, above.
Needed elements
- Platforms
- Gears
- Cylinders
How it goes
Children build a three-level construction on a platform. On the lowest level they place a cylinder on the platform, on it, in the middle — a gear, and at the very top — a second gear. Then they describe the position of the pieces relative to one another.
Variants
Change the positions and use the terms: “on the left”, “on the right”, “behind”, “in front of”.
Building a construction from a partner's instructions (“Place the cylinder under the gear”).
Tower of Gears
Learning goal
Comparing height and developing the ideas: higher, lower, at the same height.
Needed elements
- Gears
- Cylinders
- Platforms
How it goes
Children build 2–3 towers, each from five pieces of their choice (e.g. one from gears only, another from cylinders, another from gears and cylinders). Then they compare their height and order them from the shortest to the tallest.
Variants
Add measuring height (e.g. with a ruler or by counting pieces).
A task with a constraint: “Build a tower taller than 10 cm but shorter than 15 cm” (not necessarily from 5 pieces).
Obstacle Course
Learning goal
Developing spatial orientation: next to, between, behind, in front of.
Needed elements
- Various pieces (platforms, cylinders, gears)
- A figure of a person (or any small object)
How it goes
On a platform, children arrange a “course” from pieces along which a figure “walks”. Pupils describe the route using spatial terms: “walked next to the cylinder”, “stepped onto the platform”, “turned behind the gear”.
Variants
A child builds a course from the teacher's spoken instructions. In pairs: one child builds a route from a partner's instructions; the other walks it, describing how they move. Then the partner walks the same route and must describe it in a different way (e.g. child 1 says “I walked on the left side of the gear”, while the second says “I walked not on the right but on the other side—”).
Gear Treasure Map
Learning goal
Planning in space, developing directional ideas.
Needed elements
- A large platform (e.g. 9 platforms per pair, arranged in a 3×3 square)
- Gears
- Cylinders as obstacles
- A “treasure” symbol (e.g. a block of a different color or a slip of paper)
How it goes
On the platform, children place a “treasure” and obstacles. The second child must reach the treasure according to the clues, placing gears next to one another so that they turn, marking the route (e.g. “move two spaces to the right, then up, go around the cylinder on the left”).
Variants
Before children place the treasure and the starting point, they may draw a “map” on paper.
Change the positions of pieces during the game.
Prepare a slip of paper with the treasure.
Who Reaches Farther
Learning goal
Comparing length and distance.
Needed elements
- Gears
- Cylinders
- A measuring tape
How it goes
On the carpet, children build “arms” from the same number of pieces. Each block in an arm may be placed in the same way or in a different way. Then they compare which “arm” reaches farther — they may measure with a ruler or count the pieces. They develop the ideas: longer, shorter, the same.
Variants
Children themselves plan how to lengthen the arm. You can also ask for the arm to have a given length.
Add a limit on the number of blocks so that children can think for themselves how to arrange them.
Bridge
Learning goal
Developing the ideas of width, length, and height.
Needed elements
- Platforms
- Cylinders as pillars
- Gears as the roadway
How it goes
Children design and build a bridge between two points (e.g. between two platforms). Then they describe its dimensions: length, height above the ground, width.
Variants
An extension — the bridge must hold a chosen object (you can consider adding another pillar). The bridge must be wider than X cm but shorter than Y cm.
Prepare several bridge models.
Direction Puzzle
Learning goal
Consolidating the ideas: to the right, to the left, up, down.
Needed elements
- A grid board (e.g. 4×4)
- Gears
- Cylinders
How it goes
The teacher builds a 4×4 mat (from 4 platforms) and places it opposite the seated children. The teacher asks the children to raise their right hand, then after a moment to raise their left hand. Any mistakes are corrected.
Then the teacher places a red gear in the top-left corner of the mat and says that the red gear is on the left side of the mat at the top. Then the teacher places a blue gear in the bottom-right corner of the mat and says that it is at the bottom of the mat on the right side.
The teacher asks one pupil to place a yellow gear at the bottom on the left side of the mat, and another pupil to place a purple gear at the top on the right side of the mat.
After explaining the activity, each pupil builds their own mat. The teacher gives instructions: “Place a red gear in the top row in the second space from the left, then place a cylinder one space farther, to the right of the gear— and so on.” Children reproduce the arrangement.
Variants
Pupils invent tasks for one another.
Add several building levels (place a yellow cylinder on the red gear).
Cylinder in the Maze — Whole-Class Activity
Learning goal
Moving in space, directions, planning movement.
Needed elements
- Platforms
- Cylinders or gears as obstacles
- One cylinder as a “token”
How it goes
Children build a large platform, e.g. from 25 platforms (5×5), and together they arrange a simple maze (walls) from cylinders or gears, then they guide one cylinder step by step (or another “ball”), describing at the same time which way the “token” is moving.
Variants
Rebuild the maze each turn.
Platform Comparisons
Learning goal
Comparing size (length, width, height) of different constructions.
Needed elements
- Platforms
- Cylinders
- Gears
How it goes
In pairs, children build different constructions on platforms. Then they compare them by several criteria: which is taller, which is longer, which is wider at a given level (e.g. at a height of 10 cm).
Where Do I Put It?
Learning goal
Practicing reading and listening with understanding.
Needed elements
- Blocks
- Printed instructions
How it goes
The teacher divides the children into pairs; each pair gets a set and builds a 4×4 mat (a square made from four platforms). Each child in the pair receives 2 printed instructions, which they pass to the other person in the pair to carry out. It is important that the children sit next to each other, on the same side of the mat. Children take turns carrying out the instructions in order:
● Place a yellow gear at the top on the left.
● Place a red gear to the right of the yellow gear.
● Place a blue gear under the red gear.
● Place a purple gear to the left of the blue gear.
Variants
The child who reads the instructions sits with their back to the one who is arranging; only after the task is finished do they check whether the arrangement of gears on the mat matches the instructions.
My Left, My Right
Learning goal
Practicing laterality and understanding the relativity of these ideas.
Needed elements
Blocks
How it goes
The teacher asks the children to stand facing them and point to the classroom wall on their left, then after a moment to point to the wall on their right. Any mistakes are corrected.
Then the teacher asks the children to turn their backs and point to the wall on their left, and after a moment to the wall on their right. The teacher asks the children to return to their places and asks why, although the position of the walls did not change, we used different phrases to name them depending on how we were standing. The teacher explains, demonstrating on themselves, that “my left / my right” does not always mean the same direction, but depends on our position relative to one another or to an object.
To illustrate the problem more clearly, the teacher may take a blue gear in their right hand and a red gear in their left, so that the children can see the gears in their hands the whole time. The teacher says they are holding the blue gear in the right hand and the red one in the left. Turning alternately to face the pupils and then with their back to them, the teacher indicates their left and right side.
Cylinders Between the Gears
Learning goal
Practicing laterality, following spoken instructions.
Needed elements
Blocks
How it goes
The teacher divides the children into pairs; each pair gets a set of blocks and builds two 4×2 mats (two platforms joined into a rectangle). The children sit facing each other and each places gears and cylinders on their own mat as follows:
● Place a purple gear in the top-right corner.
● Place a blue gear under the purple gear.
● Place a red gear to the left of the purple gear.
● Place a yellow gear in the bottom-left corner.
● Place a purple cylinder to the right of the yellow gear.
● Place a yellow cylinder to the left of the red gear.
● Place a red cylinder in the top-left corner.
The children compare their mats while sitting opposite each other, then sit with their mats side by side and check whether the task was done correctly.
Tall, Taller, Tallest
Learning goal
Comparing and ordering by size.
Needed elements
Blocks
How it goes
The teacher builds a 4×2 mat and four towers:
● From six blue gears.
● From eight yellow gears.
● From four purple gears.
● From two red gears.
Then the teacher asks one pupil to place the towers on the mat from tallest to shortest, so that the tallest is farthest to the left and the shortest is farthest to the right of the mat.
The teacher rebuilds the towers so that:
● The first tower consists of three yellow, one red, two blue, and two purple gears.
● The second tower consists of four yellow, one purple, and one blue gear.
● The third tower consists of one yellow and one purple gear.
● The fourth tower consists of three blue and one red gear.
Then the teacher asks the pupil to place the towers on the mat so that the tower with the greatest number of yellow gears is farthest to the left, and then the towers with fewer and fewer yellow gears. At the end there should be the tower that has no yellow gears at all.
Variants
The teacher divides the pupils into teams of 4–5; each team gets 60 gears, is to build four towers of different sizes from them, and place them on platforms from largest to smallest.