A shallow understanding of math is relatively easy to attain. I memorize, for example:

7 + 6 = 13

A shallow understanding may be quickly revealed when a student is asked to solve:

y + 6 = 13

Or, (far) worse:

13 = y + 6

“I don’t get it.”

Many students who breeze through

4 + 7 = x,
3 + 9 = y
11 + 12 = z

… balk at:

a + 7 = 11
12 = b + 9
23 = 12 + c

In addition to concrete manipulative like fingers, dots on scrap paper, number lines, Rekenreks, 10-base block, tokens and money, FACT FAMILIES can help students see the relationships between the 3 numbers that make up the family.

Consider the Fact Family comprised of 3, 4 and 7.

Notice that the largest number goes in the apex of the triangle. (It doesn’t matter where the two smaller numbers go.)

Addition sentences for fact families END with the largest number
Subtraction sentence for fact families START with the largest number

While the two bullet points above are obvious to adults, they really take some thinking through for a young mathematician still more concrete than abstract in her/his thinking.

If you’re up for a fun math talk with your student, discuss this with some concrete objects on hand, wondering why this is, wondering if it’s true for all (2G-level) addition/subtraction sentences—and why.

Notice that every fact family has 4 related equations or number sentences:

2 addition sentences/equations (3 + 4 = 7 and 4 + 3 = 7)
2 subtraction sentences/equations (7 - 4 = 3 and 7 - 3 = 4)

Now, once a student “gets” fact families, s/he will start to see how useful they are. S/he can see

y + 4 = 7

as—

y + 4 = 7 is in the same fact family as 7 - 4 = y — and I can do 7 - 4 — it’s 3!

Grappling with fact families help young mathematicians develop a much more robust appreciation of how numbers in an addition or subtraction sentence — whether we know all the values, or only two out of three — relate to one another.

A real revelation: if I have the 3 numbers of a fact family, I can create the 4 equations without doing any math!

Fact families are quite wonderful, and have uses aplenty!

And, yes, one can have multiplication/division fact families such as 3, 4, 12 (3 x 4 = 12, 4 3 = 12, 12 ÷ 3 = 4, 12 ÷ 4 = 3). But that can wait until 3G and beyond!