Explore special fact families where two of the numbers are the same (e.g., 5, 5, 25). Understand why these families have only one multiplication and one division fact, instead of the usual two of each.
What is a "special" fact family, also known as a "doubles" or "square" fact family?
It is a fact family where two of the three numbers are the same because a number is multiplied by itself. For example, the family for \(4\), \(4\), \(16\).
How many unique facts (equations) are in a special fact family like \(4\), \(4\), \(16\)?
There are only two unique facts: one multiplication fact and one division fact.
Write the complete fact family for the numbers \(5\), \(5\), \(25\).
\(5 \times 5 = 25\) and \(25 \div 5 = 5\).
Why does the fact family for \(5\), \(5\), \(25\) only have two facts?
Because the factors (\(5\) and \(5\)) are the same. Reversing them in the multiplication sentence gives the exact same equation, which means there's only one unique division fact possible.
Create the fact family for \(8\), \(8\), \(64\).
\(8 \times 8 = 64\) and \(64 \div 8 = 8\).
If a fact family has only one multiplication fact, what must be true about its factors?
The two factors must be the same number.
The numbers \(7\), \(7\), \(49\) form a special fact family. What are the two facts?
\(7 \times 7 = 49\) and \(49 \div 7 = 7\).
Does the fact family for \(3\), \(6\), \(18\) have two facts or four facts? Why?
It has four facts because the factors (\(3\) and \(6\)) are different. The Commutative Property applies, giving two different multiplication and division facts.
