Today I'd like to discuss one path that takes students through "strategies based on place value and the properties of operations" in the quest to multiply multi-digit numbers by one-digit numbers. This is how it's playing out in my fourth grade classroom. (We are focusing our learning on equations instead of arrays or area models.)
While my class studied numeration, addition, and subtraction, students were also working to master multiplication facts from zero to nine. (This step is not conceptual but necessary for success. I trust that my students' third grade teachers have handled the conceptual side of single-digit multiplication.) As we move into our unit on multiplying multi-digit numbers, I am proud to report that 26 of my 27 students can fluently recall their facts. How can this be achieved? For some ideas, read my blog from August 31st.
We learned the commutative (order), associative (grouping), identity (one), and zero properties of multiplication, used the properties to determine unknowns in equations, and identified properties used. This worksheet from Scott Foresman is a simplified version of what I used in my classroom. In searching the Internet, I also found this worksheet from Common Core Sheets.
Students practiced multiplying single-digit numbers by multiples of 10 and 100. First we built it out, like this, 7 x 3 = 21, 7 x 30 = 210, and 7 x 300 = 2100. In order to maximize conceptualization, we also decomposed the numbers so students could reason why this is so: 7 x 30 = 7 x (3 x 10) and the associative property allows us to change the grouping to (7 x 3) x 10. Next students practiced using the skill with mixed problems, such as 6 x 800 = n and 40 x 3 = n. You can find some free practice worksheets at Math Worksheets Land and Common Core Sheets.
Now it's time to introduce the distributive property and employ knowledge of expanded form. Each year I worry that my students will get stumped at this stage of the game, but they always pull it off! This year was no different. Let's say, for example, we have the problem 6 x 35. We first decompose 35 to 30 + 5, so now we have 6 x (30 + 5). Then we use the distributive property to get (6 x 30) + (6 x 5). After solving for each quantity, we end up with 180 + 30. Easy! 210. You can find worksheets for practicing the distributive property at Common Core Sheets.
I often hear teachers say, "I hate partial products. What's the point?" The point is that this is the next conceptual link in the chain of events that lead to students' understanding of the multiplication algorithm. After students see how the distributive property works, they can understand that 35 x 6 = 30 + 180. The order has changed somewhat, but clearly 6 x 5 = 30 and 6 x 30 = 180. When added together, we get 210.
Finally we reach the common algorithm for multiplying a two-digit number by a one-digit number. It's the same as the partial products method, but we take a shortcut. Instead of writing 30 for the answer to 6 x 5, we simply write the number of ones (0) in the ones place and move the number of tens (3) above to be added after we've multiplied 3 tens by 6. It sounds confusing as I write it here, but the kids understand it because it's the next logical step in this conceptual progression.
While you may think this is a bunch of unnecessary hullaballoo, it's not. Sure, you could skip properties of multiplication, decomposing numbers, using the distributive property, and partial products. After all, your students can learn to use the common algorithm without it. But kids deserve more. To succeed at math in the future, they need deeper understanding of mathematical operations and how they relate to the Base-10 number system. I know I sound preachy, but I'm not the only one: the Common Core State Standards are up on the soap box with me when they say, "Multiply a whole digit number of up to four digits by a one-digit whole number . . . using strategies based on place value and the properties of operations."