# Is four times always the same as two times doubled?

One of the trickiest part of teaching multiplication deals with the distributive property. Because of this, we made the distributive property the focus of Mt. Multiplis, our multiplication app.

The Common Core State Standards expect third graders to use the distributive property when multiplying (which you and I learned in middle school): “Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56” (CCSM.3.OA.B.5). That’s one complicated equation, and it’s hard to imagine that most 9-year-olds, who are just learning how to multiply, are able to make sense of that notation. So, we set out to see if we could not only make this complex content more approachable, but also fun.

At its core, the distributive property is about groups. It connects multiplication and addition by letting you break up a multiplication problem into smaller chunks. For example, when solving 7 x 8, you can think of it as 7 groups of 8. But, how does that help you multiply? Let’s say you forgot what 7 x 8 is, but you know 5 x 8 is 40. You can then think of 7 x 8 as 5 groups of 8 plus 2 more groups of 8, that’s 40 + 16.

In Mt. Multiplis students explore this property by using groups of wooden planks to build bridges. To fill a 7 by 8 area, students drag over 5 planks of 8 and then drag over another 2 planks of 8.

The best part of playtesting has been watching kids discover the applications of the distributive property on their own. While playing Level 5, which features a lot of 11 x __ problems, one fourth grader literally said, “Aha!” as he realized for the first time, why the 11s rule works. He explained to me that the 11s facts are easy (i.e. 5 x 11 = 55, etc), but that he didn’t know why the shortcut works, until playing Mt. Multiplis. Solving 11 x 7, he dragged over 10 groups of 7 to make 70 and then dragged over 1 more 7.

Another third grader was working through several 4 x __ problems. She would consistently drag out 2 planks and then use the Double-It card to make another set of 2 planks. She first saw the problem 4 x 6, dragging out 2 six-planks to make 12 then used the Double-It card to make another group of 12. She did not fully trust that the answer was actually going to be 24, but she entered it in anyway and was surprised when it was correct. Next, she solved 4 x 3 and then 4 x 8 in the same way. After solving 4 x 8, thinking about 16 doubled, she asked me, “Does that always work…with the 4 times problems…you can just double the 2 times?” Yep, it always works. And that’s some pretty sophisticated algebraic thinking. Another way to phrase her question: Does 4y = 2y + 2y? Yes, because of the distributive property.

Kara

# What Does Mickey Mouse Have to Do with Counting?

One night before bed, in the spirit of Piaget, I decided to videotape my 2-year old daughter counting. As most parents know, counting is a fundamental part of children’s everyday mathematics experiences.

Here’s an example of my daughter counting. She’s starting to show understanding of certain aspects of counting and patterns. Pay close attention to the end of the video:

Before kids slide down a slide or begin a race, they chant, “1,2,3, go!” Kids learning to count often spend what feels like hours counting everyday objects, like fingers and toes, cheerios for breakfast or stairs they are climbing.  Children learn counting as if it were a song, a rhythmic chant.  Eventually, they begin to see patterns in the song that they can extend.

Young kids are often able to count higher than you might expect if you help them with a few key numbers, particularly the decades.  For example, a 2.5-year-old might count up until 19 and then get stuck on what comes next, but after you tell her it is 20, she can keep counting until 29.  If you tell her 30 comes next, she will keep going until 39, etc. In a past blog post, I wrote about how language makes our counting words particularly tricky for English-speaking kids, especially compared with their Asian peers. See that post here: http://bit.ly/1dnOJ4c

But knowing “the counting song” is just the tip of the iceberg in learning how to enumerate, or count objects.  Researcher Rochel Gelman from Rutgers University outlines 5 key principles that underlie the ability to enumerate.

1. The stable ordering principle: This is what most of us think of when we say, “My child knows how to count.” It refers to knowing the counting sequence (or the “counting song”). But knowing the count sequence doesn’t necessarily mean you can count objects or use the sequence in any meaningful way.
2.  The one-to-one principle: Each object to be counted gets one and only one counting word.  In other words, no double counting jelly beans and no skipping objects when you count.
3. The cardinal principle: Knowing that the last number you say when counting objects refers to the entire set of objects, not just the last object. This is one of the hardest principles for little kids to grasp and has been the subject of much research.  The classic Piagetian task testing this principle is to ask a 3-year-old to count a set of blocks.  The child may be very good at carefully counting, “1,2,3,4” as he points to each of the 4 objects; however, if you then immediately cover the objects with your hands and ask how many objects he just counted, the child doesn’t know how to answer.  He needs to count them all over again, starting from 1 because he doesn’t realize that the “4” he just said refers to the set. One way to help reinforce this idea with kids is to have them always state the cardinal value after they count, i.e. “1,2,3,4. 4 blocks.”
4. The abstraction principle: You can count just about anything (blocks, candies, ideas or all three mixed together)
5. The order-irrelevance principle:  Items can be counted in any order.  This idea can be tricky for young kids, too.  If you line up blocks in a row, many kids will think you have to start from one specific end and not from the other end or even from the middle of the line.

Here’s an example of my daughter demonstrating this principle:

Dana