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I teach 4th graders, and I don't think there is any one "best" way for learning the facts. Instead, I use several approaches, and students pick up and use the ones that work best for them. We start by figuring out which ones are usually pretty easy for "us" to learn... like the doubles (2 x __ ). 1 x __ and 0 x __ are also fairly easy, and the 5's and 10's, since students have been skip counting with them for a while already. We do look for patterns in the multiples, and students make discoveries like "every other one is odd/even," or the ones digit is always a zero or a five." We try to figure out why so we can use the notion to predict the outcome for other problems. Then I use some visual aids to help my students "see" the factors combining into products. We make sets (4 sets of 6, 6 sets of 4) and "discover" that it doesn't matter which order we use. We construct rectangular arrays, and find out that each multiplication problem looks like a rectangle. We predict the shape of the rectangle (tall and skinny, for 6x1, short and fat for 1x6), and look for other ways to make the same product (2x3). We even make patterns by drawing sets of lines that cross each other: 3x6 is three vertical lines and 6 horizontal lines. Then we count the number of intersections. We note that "special" factors will form squares, like 4x4, and 9x9. This all takes time. Some students need to have those manipulatives right in their hands, and to draw the sets of dots, or crossing lines, for 6x7. The important thing is to help students gain a sense of the products, so they will see how much more quickly the products grow, compared to the sums. Eventually the "playing" around will pay off.
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