Lines and Intersection Points
What Are Lines and Intersection Points?
In everyday life, we see that some lines are straight and others are curved. If you fold a sheet of paper in half, the crease is straight. If you stretch a piece of string, it also forms a straight line.
In mathematics, though, a line is an abstraction. A true line continues endlessly in both directions. No sheet of paper could ever hold it — you’d have to imagine it going on forever.
Straight lines can cross, just like roads or streets in a city. A house built where two streets meet belongs to both of them. When two lines cross, they share a single point — the point of intersection. To mark it, we draw a small circle or dot. Curved lines can intersect too.
Why Does It Matter for Kids?
Points, lines, and intersections are at the very beginning of spatial reasoning. In their classic study, Siegel and White described how kids build mental maps in three steps:
Landmark knowledge — knowing important “points,” like home, the playground, or grandma’s house.
Route knowledge — connecting those points into “lines,” paths from one to another.
Survey knowledge — creating a full internal “map” that shows how everything fits together.
Grown-ups go through the same process when exploring a new city. At first, only a few landmarks stand out — St. Vitus Cathedral, the Prague Town Hall, Strahov Monastery, the Old-New Synagogue, or the Dancing House. Next come the streets and bridges connecting them. Only after several visits does the full mental map of the city take shape.
In one experiment (Frank, 1987), four-year-old kids explored a 25-meter maze following simple routes. They could use a map successfully when it was aligned with their direction of movement. Older kids learned to rotate the map mentally. Landmarks placed at intersections helped them navigate, while crossings themselves proved the hardest parts. Asking questions like “How many times did we turn, and where?” made navigation easier and helped kids transfer what they learned to new situations.
Over time, the map became internalized. Kids began to “see” it in their minds and no longer needed the paper version.
The book Learning to Think Spatially highlights that spatial thinking — the ability to connect lines, directions, and shapes into one meaningful whole — supports every school subject, from drawing and reading to math and science.
Modern GIS tools make this even more vivid. When kids walk through the city and watch their dot move along the map, zoom in, or rotate the view, they literally experience the lines, angles, and intersections of streets. That kind of embodied experience turns geometry into something alive and understandable — a foundation for future success in mathematics, engineering, and science.
How Do We Teach?
The idea of a straight line grows naturally from experience. Kids might picture it as part of a pattern — a line that continues beyond what they can see.
In this task, part of the picture is missing, and kids have to imagine how the lines continue across the cutout.
Lines and intersections
Complete a pattern
Three lines forming a triangle intersect at three points. Lines can create much more complex configurations too. It’s important for kids to learn to spot all the intersection points in a drawing, since this ability will help them later in geometry.
Straight lines also appear in physics. Light travels in straight lines. If a rock stands in the way, light cannot pass through it. A set of tasks asks which objects will be in the light and which will fall into shadow, or what a dragon with narrow “tunnel” vision will be able to see.
There are also beautiful puzzles about lines and intersections. Imagine a house built at the crossing of two roads — it belongs to both. Now try this: what layouts of houses and roads meet the rule that each road should have exactly two houses on it?
You can draw two parallel roads with two houses on each — four houses in total. Or you can use just three houses if one of them stands right at the intersection.
Here’s another riddle. A room has four walls, and you need to place bowls so that each wall has exactly two bowls next to it. You can do this with four bowls by placing one at each corner, or with eight bowls by keeping them off the corners. Five, six, or seven bowls also work, depending on how many you place at the corners. If each wall should have three bowls, any number from eight to twelve will do — if they’re placed just right.
Statements about pictures
Lines and intersections
Statements about pictures
Lines and intersections
Statements about pictures
Lines and intersections
Statements about pictures
Lines and intersections
2D shapes
Statements about pictures
2D shapes
Statements about pictures
Big Ideas
Kids will keep running into lines and intersections as they go deeper into math. In geometry, lines on a plane can be parallel or intersect. In space, they can also be skew — neither meet nor lie in the same plane. Engineers and architects often build curved surfaces out of skew lines. These are straight beams that never meet, yet together they form graceful, sturdy shapes.
In algebra, a line represents a linear function, such as the path of a point moving at a steady speed. The steeper the line, the faster the motion. When two such lines cross, it means the moving points arrive at the same place at the same time. It means that they meet.
In drawing and design, parallel lines are shown as meeting on the horizon. This is called perspective, and it connects to projective geometry, computer graphics, and technologies like AR, VR, and 3D rendering.
The idea of lines and intersections also plays a key role in graph theory, which often appears in math competitions. Graphs describe road networks, electrical circuits, social connections, and even neural networks. For example, Hall’s Marriage Theorem, named after mathematician Philip Hall, explains under what conditions everyone in a community can get married based on their preferences.
Finally, in geometric optics, light rays, their reflection, and refraction are all forms of lines meeting and crossing. Lenses in glasses, microscopes, and telescopes all rely on this principle. Designers and engineers use the same ideas to calculate light and shadow in real spaces.
