For example, 3, 4, and 7 are a "fact family" because they can create the four addition/subtraction equations of
3 + 4 = 7
4 + 3 = 7
7 − 4 = 3
7 − 3 = 4
A new bit of jargon to me. But it appears to help Smiley.
Anyway, addition has several conceptual steps. Let's focus on 3 + 4 = 7 for now.
First, a kid learns that addition means counting both piles of objects. We start with a pile of three and a pile of four. If we count all the objects we get to seven. (Perhaps we merge the piles, perhaps we do not.)
Second, a kid learns that it is faster to count onward from the bigger pile. We do not need to start counting at one. We can start at four (the size of the larger pile) and then keep going for the object in the smaller pile ("five, six, seven!").
Third, a kid learns basic number-line sense. Adults do not solve 14 + 2 = 16 with mental counting. We simply know that two more than 4 is 6, and similarly two more than 14 is 16. We more or less picture the number-line and just know how nearby numbers relate.
Fourth, a kid memorizes fact families. A first-grade favorite is 7 + 4 = 11 because seven and eleven rhyme. Ideally many addition problems involving small numbers are memorized so they no longer need to be solved by counting.
Then the four previous developmental steps are applied to subtraction. Consider 7 − 4 = 3.
Fifth, we could remove four objects from a pile of seven objects, and count how many remain. (The subtraction equivalent of counting both piles of objects.)
Sixth, we could count down from seven four times.
Seventh, we could count upward from four to seven. (Seeing whether counting down or up is quicker is the subtraction equivalent of counting from the bigger pile.)
Finally, we can use number-line sense or fact family memorization get the answer without counting.
Smiley needs to memorize some fact families. Flash cards are boring and not fun.
I designed something better. Reloading that page generates a new worksheet of sixteen problems.
Because of how the fact families are carefully "hidden" on the page, solving these problems is fun for him because Smiley feels like he is "cheating" by finding the patterns.
Actually, he is learning the valuable skill of not solving math problems in order, but looking for how to do the fast/related problems together.
I brought some samples to all the Edgewood first grade classrooms last week. The kids worked in pairs. The worksheets were a big success.