In Book 5: Teaching Addition, Subtraction, and Place Value, page 40, the following comment about the equals sign appears:

“Many students regard the ‘=’ sign as meaning only ‘get the answer’. That is what ‘equals’ means on a calculator. But the notion of equals is more general than this. In the problem below, the equals sign means that the total on the left of the sign is the same as the total on the right.”

This more general meaning of equals is extended into multiplication and division.

Using Number Properties

Problem: “Jocelyn knows 12 x 12 is 144, and she knows 2 is a factor of 144 because 144 is even.

Complete this statement: 12 x 12 = 2 x ?”

Discuss the balance idea of equals; the left-hand side is 144, so the right-hand side is 144.

And 2 x 72 = 144, so ? is 72.

Examples: Complete these equations: 6 x 10 = 3 x  ?    ? x 20 = 8 x 10

60 ÷ ? = 20,   4 ÷ 4 = 30 ÷ 2,   6  x ? = 3 x  40,  6 x 10 = 3 x ?,  
2 x  4 + 3 = ? + 5,  2 x  4 + ?  = 11,  2 x ? = 11 – 3,  ? – 3 = 11 + 1

Understanding Number Properties:

Find two numbers that go in the boxes to make this equation correct: 35 ÷ 5 = ? ÷ ?