## Monday, January 7, 2013

### Distributive Property=Fact Fluency

When I think back to the multiplication facts that were really hard for me to remember, the 7s, 8s, and 9s come to mind.  Of course that was back before research showed us that using strategies to learn basic facts was a much better route than drill and kill.  The more flexible students are with numbers, the easier fact fluency will come for them.  That's one reason I love Number Talks.

Apply properties of operations as strategies to multiply and divide.  Examples: If 6 x 4 = 24 is known, then 4 x 6 = 24 is also known. (Commutative property of multiplication.) 3 x 5 x 2 can be found by 3 x 5 = 15, then 15 x 2 = 30, or by 5 x 2 = 10, then 3 x 10 = 30. (Associative property of multiplication.)  Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) = (8 x 2) = 40 + 16 = 56. (Distributive property.)
Well, that's certainly a mouthful!  Just like we teach our children to break complicated problems into smaller chunks, let's break this down and look just at the distributive property tonight.
Knowing that 8 x 5 = 40 and 8 x 2 = 16, one can find 8 x 7 as 8 x (5 + 2) = (8 x 5) = (8 x 2) = 40 + 16 = 56. (Distributive property.)
8 x 7 is what might be considered a "hard" fact to learn.  But most kids easily learn their 5s. The example shown above capitalizes on the easier 5s facts by decomposing the second factor, 7, into 5 + 2.  Still pretty abstract, right?  And it will be for your kiddos, too.  That's why they need to experience the distributive property using manipulatives.

Look at this example of one way the distributive property could be used to find 4 x 7:

Remember, it is always better to let the kids learn instead of you teaching, so the secret is to let the kiddos discover the distributive property on their own!  I might try this:
Teacher: Using your counters, build an array that is 4 rows of 7.  Great!  Now, what would the number sentence be for the array you've built?  (4 x 7 = £).  Hmmmm, I don't know 4 x 7, and I don't want to count all the counters.  I wonder if there is a smaller fact in this array that I do know.  See if you can split your array into two arrays.  Don't forget to write the number sentences to go with your arrays.  Give students a few minutes to work with a partner--understand that they will probably come up with different ways. For example, one pair of students might split the 4 and make 2 rows of 7 and another 2 rows of 7, or (2 x 7) + (2 x 7).  Another pair might split the 7 into 3 + 4, giving them (4 x 3) + (4 x 4).  That's when math gets FUN!!  An idea for sharing might be to have them draw their ways and share them on a document camera. Think of the awesome conversations you'll have!  Now, you will have to show them the format for writing their equation.  For example, how to use the parenthesis and + sign.  No biggie.
I've provided a couple of other examples, but you can make up your own.  Keep in mind that students don't need to know the names of the properties. It's really the understanding of how mathematicians use the properties. I'd love to hear your comments about your experiences with this or the other properties.  Please comment!

1. Donna,
Just wanted to thank you for always posting such great information, help, and activities!
Shawna
The Picture Book Teacher's Edition

1. Aw, thanks, Shawna! You filled my bucket. :)

2. Hi,

Great to see a blog about Maths in Australia. I have written an app on Android called 'Maths Bug' that will probably help your readers improve their mental maths ability; available for free. Covers addition/subtraction, multiplication/division all the way up to fractions, angles and algebra.

Keep up the good work.

3. I am so psyched. I am right in the middle of teaching division and I had reread your blog about the distributive property and then you come out with even more help. Thank you so much!!!

1. Awesome, Nancy! Glad the timing worked out so well for you! :)