Kitchen > PrivateWebHome > WebLog > NumberBondsVersusFourFactFamilies
09 Oct 2005 - 20:26

## number bonds vs fact families

From the Comment thread about Lone Ranger's approach to teaching an 8-year old why it's OK to write the number 5 as 5/1: I mentioned that Saxon Math uses four-fact families to teach the operations of arithmetic, while both constructivist curricula and Singapore math seem to use 'number bonds.'

Here's an example of a number bond flash card:

You can download these cards from DonnaYoung.org, a homeschooling resource that looks pretty good, and has a page of mostly terrific paper math manipulatives, including lots of circular fractions, terrific large-print math facts drill sheets, graph paper, play money, scale paper for household furniture arrangements, and some cool-looking empty worksheets with number lines on top.

It also has triangular addition and subtraction flash cards (pdf file).

from the directions:

To use the cards, hide one of the corner numbers with your thumb or finger and let the child tell you what the hidden number is.

### Saxon's fact families

Saxon Math does not use triangular flash cards.

Saxon uses four-fact families combined with Extreme Practice. If there is One Thing Christopher & I have overlearned from Saxon 6/5, it is FOUR FACT FAMILIES:

1, 2, 3

1 + 2 = 3
2 + 1 = 3
3 - 2 = 1
3 - 1 = 2

Same deal with multiplication and division.

Here's a typical four-fact family problem from Lesson 2 in Saxon 7/6:

23.
Rearrange the numbers in this addition fact to form another addition fact and two subtraction facts.
12 + 24 = 36

Christopher can do that in his sleep.

So can I.

I probably have done it in my sleep.

I've been doing so much grade school math I sometimes dream about it.

### four weeks into Saxon 6/5

Quoting from a post I wrote on this subject awhile back:

About a month after Christopher and I began working with Saxon Math 6/5, he told me,

Multiplication and division are the big brothers, and addition and subtraction are the little brothers.

Then he said,

And multiplication and division are cousins.

This is a 9-year who, just 6 weeks earlier, had been flunking math.

You have to do a lot of four-fact fact families to come up with a thing like that.

### I vote for fact families

Triangular flash cards and number bonds are everywhere these days, but I don't like them. Here's why:

• First of all, the potential for confusion is huge. An addition & subtraction number card looks extremely similar to a multiplication & division number card, and separating factors from addends in a child's mind is a challenge under any circumstances.

• Second, triangular number bond cards aren't all that easy to 'read.' Kids don't naturally undestand visual displays of data; far from it. There's too much info on these cards, IMO.

• Third, number bonds are incredibly static, and I don't think math is static. Math is something you do, not something you look at. Four-fact families are action-packed; you get so good at them you can whip one of those babies out in a couple seconds flat. They're fun, and they absolutely (I'd bet money on it) prepare kids for the time when they're going to start solving problems like 2 + a = 5. When Christopher segued to 2 + a = 5 in Saxon 7/6 he didn't have a second's difficulty. He'd been inverse-operationing 2 + 3 for a year at that point, so 2 + a was just obvious.

• Last & certainly not least, I haven't had any luck with flash cards, period.

Not nearly as beautiful as Doug's number lines, but a good idea.

### oops

I've just noticed that Donna Young prefers sites not link to her printable forms, and in fact these links won't access the forms. Just go to her homepage, click on math, and then find what you're interested in. The math page is clear & easy to use.

Curricular Game Playing
Curricular Game Playing, part 2
number bonds vs. 4-fact families
Numicom Dominoes

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Great website! That's a keeper.

I agree about the fact families. I never would have thought so, but after watching my son do them ad nauseam during almost every set, it's easier to use the inverse of something as a reminder teacher tool when he gets stuck (which is alot.) It just quietly drums it in. It seems so obvious to me, but as an adult I think we just sometimes forget what they are really thinking.

-- SusanS - 09 Oct 2005

It's one of those things you have to see and/or do to believe.

I couldn't forget that multiplication and division are inverse operations now if I wanted to.

-- CatherineJohnson - 09 Oct 2005

##### quietly drums it in

Absolutely.

-- CatherineJohnson - 09 Oct 2005

Plus, I was thinking today about the whole procedural knowledge business.......procedural knowledge is usually defined in terms of motor skills. Baseball, riding a bike, etc.

Well, you know how Carolyn's always talking about the 'craft of math,' and getting math 'into' Ben's hands....that was one of her very first posts: Swoop and Swoop.

I'm thinking she's using the classic form of procedural memory there, the motor form.

My feeling about math is that it's so hard to remember, you ought to use everything you can.

So why not enlist procedural memory in the sense of motor memory??

Doing a gazillion pencil-and-paper four-fact fact families puts inverse operations in your hand.

Staring at triangular number bonds is just weird.

Plus when you write in the operation signs, the things look like something from Battlestar Galactica.

-- CatherineJohnson - 09 Oct 2005

Here's Carolyn's post on the craft of math.

-- CatherineJohnson - 09 Oct 2005

And........I'd put money on it that the normal transition from:

2 + 3 = 5

to

2 + a = 5

is not that smooth for most kids.

For Christopher it was nothing. And he has to work to learn math; he's definitely not a kid who can learn it 'by smell.'

I'm repeating myself!

But it's pretty amazing. (I guess what I mean is: it's pretty amazing when you see something go easily. Not too much does.)

-- CatherineJohnson - 09 Oct 2005