Long before writing, people kept track of herds, harvests, and trade by making marks — tallies — one mark for each item. Archaeologists have found tally bones tens of thousands of years old.
Different cultures invented symbols for amounts. The idea of zero as a number took time: Indian mathematicians such as Brahmagupta (7th century CE) wrote rules for arithmetic that included zero. The Maya used a shell sign for zero in their calendar. Zero let people write “none” clearly and build place-value systems later on.
Multiplication grew from repeated addition in trade and land measurement — counting equal rows of goods or fields without adding one row at a time forever.
Division appears in sharing problems as old as fair trade: split grain, land, or money into equal parts. Large-number names (million, billion) expanded as empires, astronomy, and finance needed to record bigger totals than everyday counting.
Suppose the first group has \(a\) objects and the second has \(b\) objects, with no overlap. If you put them in one pile and count from the start, you say \(a\) number words for the first group, then continue with \(b\) more words. The last word is the total \(a + b\). That is why \(3 + 4 = 7\): three objects, then four more, gives seven in all.
If there are \(r\) equal rows and each row has \(c\) dots, you can add \(c + c + \cdots + c\) (\(r\) times) to get the total. Writing \(r \times c\) saves repeating the same addition. For example, \(4 \times 3 = 3 + 3 + 3 + 3 = 12\).
If \(T\) dots are split into \(g\) equal piles with nothing left over, each pile has \(T \div g\) dots. That is the same as asking: what number times \(g\) gives \(T\)? So \(T \div g = q\) when \(g \times q = T\). Example: \(12 \div 4 = 3\) because \(4 \times 3 = 12\).
In base ten, each column is worth ten times the column to its right. So \(47 = 4 \times 10 + 7\). Extending the pattern: \(1{,}000 = 10^3\), \(1{,}000{,}000 = 10^6\), and each “thousand step” multiplies by \(1{,}000\) again. That is why we group digits in threes when writing millions and billions.
Pause and answer in your head or with a partner before moving on.
Checkpoint
Show the number \(7\) on your fingers. What digit symbol matches seven?
Checkpoint
Which is greater: \(5\) or \(9\)? How do you know?
Checkpoint
Is \(2 + 2 + 2\) the same total as \(3 \times 2\)? Explain with a picture of dots.
Checkpoint
What comes next in the pattern \(2, 4, 6, 8, \ldots\)?
Checkpoint
Share 15 dots into 3 equal piles. How many in each pile? What if you try 4 piles?
Checkpoint
Which is bigger: 1 million or 1 thousand? How do you know?
Answer: \(5\).
Answer: \(0\) (zero).
Answer: \(7\).
Steps: Two objects plus five more gives seven.
Answer: \(10\).
Answer: \(12\).
Steps: \(4 + 4 + 4 = 12\), or four rows of three dots.
Answer: \(8\) is larger.
Answer: \(10\).
Steps: \(5 + 3 = 8\), then \(8 + 2 = 10\).
Answer: \(9\).
Steps: \(3 \times 3 = 9\).
Answer: \(10\).
Explanation: Addition of whole numbers can be done in either order (commutative property).
Answer: \(10\).
Steps: \(2 + 2 + 2 + 2 + 2 = 10\); five rows of two dots (or two columns of five).
Answer: \(3\) per pile.
Steps: \(12 \div 4 = 3\) because \(4 \times 3 = 12\).
Answer: Six zeros: 1,000,000.