Hundreds chart

A hundreds chart is the first hundred counting numbers written ten to a row, and its usefulness comes from that one decision. Because the rows are as long as the base, moving down a row always adds ten and moving across always adds one, so the chart turns place value into something a child can point at: every number in a column ends in the same digit, and every number in a row of the 0–99 version starts with the same one.

The grid has an ancient ally in the sieve of Eratosthenes. The Greek scholar, who lived from about 276 to 194 BCE and is remembered mostly for measuring the Earth, gave his name to the method of finding primes by striking out the multiples of each prime in turn. Nicomachus of Gerasa described it around 100 CE, and a hundreds chart is still where most pupils first carry it out: cross out every second number after 2, every third after 3, and so on, and the twenty-five primes below 100 are what remains.

The chart's companion story is about adding it all up. Wolfgang Sartorius von Waltershausen, writing in 1856, recalled that the young Carl Friedrich Gauss was set to add the numbers 1 to 100 and handed in the answer almost at once, by pairing 1 with 100, 2 with 99, and so on — fifty pairs of 101, which is 5,050.

Modern classrooms use the chart for counting on and back, skip counting, and spotting patterns. Montessori schools use a wooden hundred board with numbered tiles. The 120 chart exists because the Common Core standards adopted across much of the United States from 2010 ask first-graders to count to 120, starting at any number less than 120.

Key properties

• There are 25 prime numbers between 1 and 100, from 2 to 97.
• There are 10 square numbers in a 1–100 chart, from 1 to 100, and the gaps between them are the odd numbers 3, 5, 7 and so on.
• The numbers from 1 to 100 add up to 5,050, which is 100 × 101 ÷ 2.
• Moving one cell down adds 10, moving one cell right adds 1, so a step down and to the left adds 9 and a step down and to the right adds 11.
• That is why the multiples of 9 form a diagonal running down to the left, and the multiples of 11 a diagonal running down to the right.
• The multiples of 2, 5 and 10 stand in whole columns, because each of those numbers divides 10 exactly.
• In the 0–99 chart, the row is the tens digit and the column is the ones digit, so every two-digit number sits at its own coordinates.

Where they turn up

• Primary-school maths lessons, from counting by tens in kindergarten to the sieve of Eratosthenes a few years later.
• The "100th day of school", widely celebrated in American elementary schools, which classes often count off on a hundreds chart.
• Missing-number worksheets, where pupils fill the blanks by reasoning from the numbers around them.
• Montessori hundred boards, where children lay numbered tiles into a ten-by-ten frame.
• Puzzle books and number games that hide a picture in the chart, revealed by shading a list of numbers.