Symmetry
What Is Symmetry?
Symmetry means seeing a perfect copy of a shape or pattern on the other side. Imagine folding a piece of paper in half, cutting out half a heart, and then opening it to see the full shape. The complete heart that appears is perfectly symmetrical.
The same idea shows up when kids make paper snowflakes. They fold the paper several times, cut out tiny shapes, and then unfold it to find a snowflake with many lines of symmetry. It is always a small surprise because every fold and cut creates something new.
With just a few folds, you can even make a surprisingly complex design using only one cut.
Why Does It Matter for Kids?
Understanding symmetry is one of the big milestones in early math. The National Council of Teachers of Mathematics highlights it as a key goal for elementary school. When kids learn to notice symmetry and use it to describe shapes and patterns, they are also building skills that connect to counting, algebra, and measurement.
Studies show that spatial reasoning and math understanding grow hand in hand. Symmetry plays a big part in this connection. When kids see that the number line is symmetrical around zero and try folding it in their minds, they start to understand how negative numbers work. This way of thinking later helps them with fractions, equations, and finding missing numbers.
Visual and digital artefacts, when combined with thoughtful teacher guidance, can greatly enhance kids’ understanding of symmetry and the related mathematical ideas.
How Do We Teach?
Kids start by spotting symmetry in everyday things. They look for pictures or objects that have matching sides, or find one that looks like the mirror image of another. Then they move on to finding the missing “half” that completes a symmetrical picture. In reflection games, kids compare real objects with their mirror or water reflections and decide which ones look right and which ones don’t.
On the Geoboard, kids stretch colorful rubber bands across a grid of pegs to make symmetrical designs on the screen. At first, the line of symmetry is vertical or horizontal; later, it can be diagonal. As kids gain confidence, the patterns become more intricate and exciting.
Other activities ask kids to imagine what will happen when a folded paper with cut-out shapes is unfolded. They try to picture the final result in their minds — a great way to strengthen visual reasoning.
Later come patterns with more than one line of symmetry, like snowflakes, stars, or spider webs. A special craft mission explores how to fold a sheet of paper so that one single cut can make a complex figure, such as a heart or a tree.
First Steps
Paired and Symmetrical Objects
Young kids start by matching pairs of familiar things — socks, mittens, or gloves. Some pairs are exactly the same, while others are mirror images. For example, the left shoe looks almost like the right one, but you can’t wear it on the other foot.
Symmetry
Symmetry in pictures
Step by step, kids notice that some objects are symmetrical and some are not. Braiding two pigtails makes a symmetrical hairstyle, but one braid on one side does not. A mushroom looks the same from all directions, while a mitten has a special “thumb house” only on one side.
If you draw two right wings of a butterfly on a sheet of paper and then fold it so the wings press against the blank side, you’ll get a full butterfly when you unfold it. The wings will match perfectly.
Kids also solve puzzles where they have to find the missing half of a cookie, butterfly, or another symmetrical shape.
Symmetry
Reflection symmetry
Symmetry
Symmetry in pictures
Symmetry
Symmetry in pictures
Symmetry
Reflection symmetry
Deep Understanding
Mirror Reflection
When something is reflected in water, the duck or the landscape should look the same as the real one, only turned upside down. Kids get tasks where they need to spot which reflections are correct and which are not.
At first, the exercises are simple — for example, the reflection might show a duck of the wrong color. Later, the challenges get harder, like finding the missing car in a mirror image.
Symmetry
Symmetry in pictures
In a mirror reflection, a person standing with their back to us would see their own face. An object held in their left hand still appears on the left side of the mirror, which means the reflection seems to hold it in its right hand.
The same thing happens when we look at a number from the other side — for example, a birthday cake with a “7” looks as if the number were flipped in a mirror.
When a sheet of paper is folded along a line, the folded part becomes a mirror image of the other side. In the mental folding challenge, kids see a shape with unfolded edges and imagine what it will look like once those edges are folded in. To find the right answer, they have to visualize the folding process and picture the final shape in their minds.
Symmetry
Reflections
Left & right
Explore mirror reflections
Symmetry
Reflection symmetry
Mental folding
Mental folding: two or three parts to fold
Confident Mastery
Make It Symmetrical
On the Geoboard, kids stretch rubber bands across pegs to make symmetrical figures. The first levels are simple: the line of symmetry is horizontal, and the shape might be something like a diamond. Later, the line can run diagonally, and the figures become more intricate and detailed. In the most advanced tasks, kids work with several rubber bands at once to create layered symmetrical designs.
Symmetry challenge
Vertical, symmetry, 2–3 polygonal lines of 3–7 segments
In cutting activities, kids imagine what will happen when a folded piece of paper is cut and then opened. The result is a shape with two or more lines of symmetry. In some challenges, the goal is to figure out how many lines of symmetry a figure has.
In a special craft mission, kids learn how to fold a sheet of paper so that one single cut can create a complex design, such as a heart or a tree. Parents and kids can try these “one-cut” tricks together at home or invent their own.
Reflection symmetry
Fold and cut
Symmetry
Make a shape symmetrical
Symmetry
Reflection symmetry
Symmetry
Math fun with scissors
Big Ideas
The idea of symmetry goes far beyond paper crafts. It plays a major role in geometry, algebra, chemistry, and physics.
In geometry, symmetry often leads to elegant and unexpected solutions. Algebra explores transformations that map a figure onto itself. For example, the group of symmetries of a square has eight elements — reflections across its midlines and diagonals, along with rotations. Mathematicians have classified all types of motions in the plane and in three-dimensional space, including reflections, rotations, and translations. This same theory also helps describe how patterns repeat in tilings and crystal structures.
Crystallography studies the symmetry of atomic lattices. For instance, frozen water molecules form a cubic lattice, while carbon atoms in a diamond create a much denser structure. The properties of these lattices determine how the materials behave — that is why diamonds are so incredibly hard.
In organic chemistry, symmetry appears again in a surprising way. Molecules in nature are not always symmetrical. Take glucose, for example. Our bodies easily digest D-glucose, but its mirror image, L-glucose, cannot be processed and simply passes through the body unchanged.
Symmetry is not just about shapes and patterns — it is one of the deepest ideas linking math, science, and the world around us.
