I thought people might like to have the latter set of worksheets, so I'm posting them here (PDF).

- The first page is a skills brush-up.
- She then filled out the second page, while I co-piloted as necessary.
- Finally she completed page 3, which summarizes the conceptual work of page 2.

The last question on page 3 is an unrelated question about perfect numbers. At dinner the night before, I had remarked that 6 is called "perfect," because if you add up the factors of 6 not counting 6 itself, you get 1 + 2 + 3 = 6. Perfect numbers are fun (here is some cool background on them) but my underlying purpose in asking about them was actually to provide a bit of brush-up with multiplication facts.

Getting back to the worksheets, you'll see on page 2 that I use a visual model to make the distributive property concrete. However, there are some differences between page 2 and what one often sees in math textbooks. One difference is that I'm not referring anywhere to area. This isn't facially an area model. Instead, we are counting things. It's true that the things being counted are squares, so that if we conceive of the squares as area units, then we are also finding the total area of the strip. And as we get closer to the time of multiplying fractions, I'll be casting the squares as area units. But right now, I'm not introducing that language or activating those concepts in the student's mind. I'm staying close to the elementary meaning of multiplication, which is to count things arrayed in groups. (I did choose squares with an eye toward area models and fraction operations later on; thus, the diagram isn't baskets of flowers or the like.)

The second difference is that the diagrams on page 2 are numerically accurate. I didn't merely gesture at the ideas by drawing rectangular fields of color with arbitrary dimensions. In these diagrams, you can actually count all of the squares, and if you do then you will get the right answer.

I think that in the early stages of this kind of learning, textbooks often assume too much about what students do and don't understand about the visual representations the textbook uses. The author understands the elasticity of the representation scheme, but I doubt that all of the students do. To motivate multiplication as a better alternative to counting, let's be sure that multiplying and counting give the same answer! And in a case like this worksheet, where I'm using the distributive property to structure the multiplication into easy steps, I want each of those steps to give the right answer too. I'm suspicious of representations that ask beginners to think metaphorically.

Finally, in page 3 I'm making sure to summarize the visual/concrete work symbolically. Symbolic calculations like those on page 3 are going to generalize not only to fractions, but also to variables, expressions, area integrals, quantum-mechanical expectation values, you name it.

What's next? For the younger, we'll start practicing easy right-to-left vertical additions (no regrouping, or regrouping in one place). And I want to make sure there are no cobwebs about place value, by using the excellent place-value mini-assessment on achievethecore.org. For the elder, we'll begin building fluency with multiplying a multi-digit whole number by a one-digit whole number. As that takes hold, we can begin stacking those calculations per the standard multiplication algorithm.

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