Use hieroglyphics for all numbers. The units are written with a single stroke, the tens are written with an inverted U, the hundreds are written with a coil, the thousands are written with a lotus plant, the ten-thousands with a finger, the hundred-thousands with a toad and the millions with a person with upraised arms. The number 147 would be written with one coil, four inverted U's and seven strokes. Addition and subtraction were very similar to the way we do them today. Multiplication and division were much more complicated.
Make fractions by adding inverses of numbers. This means that all Egyptian fractions had 1 in the numerator or were the sum of a series of fractions with 1 in the numerator. This was the only approach to fractions used in western civilization until the 17th century. Egyptians never had a proof that every fraction could be written this way; the proof did not come until 1880. Although the Greeks named the symbol "pi," they got the idea from the Egyptians, who represented it as 3 + 1/13 + 1/17 + 1/173. This comes out to 3.141527, which is very close to pi's actual value of 3.141592.
Get the laws and relationships of geometry from a collection of individual problems. Every year when the Nile flooded, property boundaries had to be redrawn. One of the devices that Egyptians used for this purpose was three wooden stakes connected by rope. The lengths of the ropes were in the ratio 3-4-5. When the stakes were driven into the ground with the ropes stretched tight, the device marked out a 90-degree angle. Geometry was a practical issue for the Egyptians, yet they knew things like the volume of a square pyramid: V = (h/3)a^2, where h is the height and a is the length of one side.