Fun with fractions

You know what fact families are, right?

They are sets of related number facts. 5+3=8, so also 8-5=3 and 8-3=5. Some people get very worked up about whether to include the commutative fact 3+5=8 in the family. But we are not worried about that here.

In elementary curriculum, fact families are used to help students learn their arithmetic facts and to highlight important inverse relationships among operations. So we use addition and subtraction to make fact families and we use multiplication and division to make fact families.

A colleague sent this idea along for a task:

I got to wondering whether this task is ever ambiguous.

Are there any sets of three numbers where both addition/subtraction and multiplication/division can be used to create a fact family?

2, 2, 4 is a well-known case, but I haven’t thought of it this way before. 2+2=4, so we begin the addition/subtraction fact family. But also 2*2=4, so we get a multiplication/division fact family.

It turns out that if we allow fractions, there are lots of possibilities.

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