Place Value, Fractions, Nets and Euler's Formula

Place value: blocks that build a number

Our number system is built on groups of ten. Picture a small cube as 1. Ten of them in a row make a rod worth 10. Ten rods side by side make a flat worth 100. Ten flats stacked make a large cube worth 1,000.

To read a pile of blocks, count each kind and add. Three flats, five rods and two small cubes make 300 + 50 + 2 = 352. The position of a digit tells you which kind of block it counts, which is why 352 and 523 are different numbers made from the same digits.

Trading ten for one

Ten blocks of one size are worth exactly one block of the next size up. So you can trade ten small cubes for one rod, and the number does not change. You can also break a rod back into ten small cubes. This is what you are really doing when you carry and borrow in written addition and subtraction: regrouping, not magic.

Fractions are fair shares

A fraction describes a share of one whole. Its denominator (the lower figure) names the count of equal pieces the whole was split into, and its numerator (the upper figure) counts the pieces you hold. So 3/8 means splitting a whole into eight equal pieces and holding three.

The word equal matters. A chocolate bar broken into one huge piece and two crumbs has not been divided into thirds.

Different-looking fractions can be the same amount. Cut every slice of a half-coloured pizza in two and you get 2/4 coloured, then 4/8, and nobody added any pizza. These are equivalent fractions. To compare two fractions of the same whole, first check the bottoms: if they match, the bigger top is bigger; if they do not, rename them as equivalent fractions with the same bottom.

Faces, edges and vertices

Every solid with flat sides has three things you can count:

• faces: each flat surface of the solid;
• edges: every straight line where a pair of faces join;
• vertices: the pointed corners, where several edges come together.

A cube has 6 faces, 12 edges and 8 vertices. A square-based pyramid has 5 faces, 8 edges and 5 vertices.

Now slit a cereal box along a few of its edges and flatten it on the table. The flat pattern you get is called a net. A net is a recipe: fold it along the creases and it closes back into the solid. Looking at a net is a good way to count faces and to see which faces touch.

Which patterns fold into a cube?

Join six squares edge to edge and you get a hexomino. There are 35 different ones, but only 11 fold into a cube. Flipping or turning a pattern does not count as a new one.

Many of the working nets share a shape. A strip of four squares wraps round the cube as a belt, and the other two squares are the lid and the base, attached anywhere along the belt on opposite sides.

Why do the rest fail? Two squares land on the same face and a different face is left open. A 2 × 2 block of squares can never work, because only three faces meet at any corner of a cube, never four. A row of five or six squares also fails, since it wraps past the start and overlaps itself. Seeing why a net fails is the real skill, and it trains your spatial reasoning.

Euler's formula: F − E + V = 2

Count the faces (F), edges (E) and vertices (V) of a solid with flat faces and no holes through it, then work out F − E + V. The answer is always 2.

• A cube gives 6 − 12 + 8 = 2.
• A triangular prism gives 5 − 9 + 6 = 2.
• An octahedron gives 8 − 12 + 6 = 2.

Leonhard Euler wrote this relationship down in the 1750s, and it is one of the first results in topology, the study of shape that ignores exact size and angle. The surprise is that the answer does not depend on how the solid is built. Squash or stretch it and the counts stay the same.

Holes change the rule. A picture frame shape with a hole through it gives a different total, which is a clue that the 2 reflects the overall shape of the surface, not the details.

Common mistakes to avoid

• Reading digits instead of positions. The 2 in 207 is worth 200, not 2.
• Comparing fractions by their tops alone. 1/8 is smaller than 1/4, even though both have a 1.
• Counting hidden edges twice. Each edge is shared by two faces, so count it once.
• Assuming any six squares make a cube net. Only 11 of the 35 patterns do.

How to explore this in the lab

The Shape Lab turns these ideas into things you can fold, count and build. Its free first worlds follow this order:

1. Build and read numbers in Ones, tens and hundreds.
2. Cut a whole into equal parts in Fractions are fair shares.
3. Count the parts of a solid in Faces, edges and vertices.
4. Fold the patterns in The 11 nets of a cube.
5. Test the rule yourself in Euler's formula.

The lab is a simplified teaching model with ideal solids, so use it to build intuition and then try the same counts on real boxes at home.