# Number Choices Matter

I had the privilege to work with fourth and fifth grade teachers this week. We explored multiplicative comparison problems in fourth and division with fifth (more on those in another blog). What I came away with is this: NUMBER CHOICES MATTER.

We don’t get much choice on the concepts we teach, nor often on the program we have to use. But we do get to choose what numbers we use with children. This could make all the difference for kiddos. If we choose our numbers wisely, we can build understanding through the patterns they see, the differences that appear, and talk about why those differences happened.

For example, consider the following set of problems: What is the same about each? If I am thinking about this as a partitive model and using base 10 blocks, I am sorting the amount I am given into 3 equal groups each time.

What is different about each? The amount I am giving to each of the 3 groups.

In the first example, 36 can be created by using 3-tens and 6-ones, and each can be fairly shared without any problems. Each would get a ten and two ones, or 12.

In the second example, I can still give out a ten to each of the 3 groups, but now I have to figure out what to do with the leftover ten and eight ones. This one builds off the first example, but pushes students to think about exchanging (regrouping).

The final example builds off the 48, but leaves 2 left that students have to consider. This allows to have a conversation about remainders.

These three build understanding of division, regrouping and remainders through strategically chosen problems to build from one to the next. Students have something to grasp on to when negotiating meaning with this tough tough subject.

So where can you build understanding through your number choices? I challenge you to think about what you want your students to learn next week and how your number choices can contribute to students understanding those goals!!!!

# Why Distribute in Third Grade?

I am blessed to work with dedicated teachers who care deeply for their students and are working hard to understand the conceptual shifts CCSS brings to the table in math. One such teacher emailed me this weekend distraught, not knowing how to respond to a frustrated parent. The premise was the mother did not understand why her third grader was being asked to learn the distributive property, when she herself hadn’t learned it until Algebra I.

The arithmetic properties (commutative, associative, identities, etc.) were not created for Algebra I, though many of us didn’t learn them until then. I remember thinking that a bunch of old guys must have made them up for the sheer joy of torturing me into memorizing random stuff. No, the properties are the rules that give us the freedom to simplify math problems to make them easier to calculate while keeping the value the same.

An example. Which would be easier to solve in your head, 15 + 29 then add 5  OR  15 + 5 then add 29? The second, because we can make tens (15+5 = 20) and easily add 29 to it rather than have to “carry the 1” on the first example. This illustrates the commutative property: when adding or multiplying, I can perform that operation with any numbers in the problem first. I can switch the numbers around to make the problem easier to add (or multiply).

These properties should be celebrated as early as Kindergarten. Students do not necessarily need to know the names, but should realize through exploration that they exist and help them find their values.

Back to the distributive property. We old-timers saw it used like this: 6(5x + 2) = 30x + 12. This is not what we are asking third graders to do! Since the CCSSM standards require single digit multiplication fluency in third grade (1×1 through 9×9), it is natural to teach the distributive property at this level. This property allows me to break up one of my bigger numbers into parts. I can then multiply those smaller parts by the other factor to make it easier. Example: 6 x 7   This is always a toughie. Is it 48? 42? 56? I don’t know! Even if I draw an array (the above pic), that is a whole lot of dots to count!

But if I know my 5’s and 2’s, I can figure it out using the distributive property!             Break up the 7 into 5 and 2.   So now 6 x 7 becomes 6 x (5 + 2). I know 6 x 5 = 30.  (This is illustrated in yellow.)

I know 6 x 2 = 12 (This is illustrated in red.)

So 6 x (5 + 2) = 30 + 12 = 42. You may also see the distributive property used as a “number bond”. This is when it is broken up into parts, either using boxes or circles to show the parts. In this case, 7 is broken up into two parts:  5 and 2.

Utilizing the distributive property is an amazing strategy for students who struggle with memorization! They can use the distributive property to break up larger values they don’t know the multiplication facts for, until they have time to build fluency. More important, it teaches kids the value of knowing that math isn’t just memorization. There are structures and patterns that I can use. If I know the rules that govern those patterns (the properties), I can change the structure to find the value in an easier way.