One of the things I feel passionate about is teaching math facts strategically even with our youngest students. Not only are we teaching them strategies for figuring out the particular math facts that are appropriate for their development, but there are two additional bonuses: we are teaching about the conceptual meaning of the operation itself AND we are providing our students with the confidence to try to solve problems they never thought they could (as opposed to students who have learned by memorizing math facts and won’t even attempt more difficult ones because they say they “haven’t learned that one yet”).
Recently I worked with some students to understand the strategy of double, double, double with their multiplication math facts of 8. One of my favorite and, I believe, most powerful math tools is Cuisenaire rods because they help children see that numbers exist as a group rather than a collection ones that need to be counted. I particularly love them to show multiplication problems. We first began by building our rectangle for the problem 8 x 6. My students build their rectangles with the vertical alignment of the first factor and then the horizontal alignment of the second factor (this helps tremendously down the road when we do division). Here’s a picture of it:
There were 8 of the 6 rods, so we built an 8 by 6 rectangle and we then needed to figure out the area inside the rectangle. I wanted to help students find an efficient way for figuring this out rather than just skip counting by six 8 times. So, I had them separate the rods like this:
That way, they could see that the 8 x 6 rectangle has the same area as two 4 x 6 rectangles. We discussed that we can think of any number times 8 as being the same product as a doubling of 4 times that number.
There were a couple of students in my small group who didn’t know that 4 x 6 was 24 so I had them separate their rods like this:
They could see that x4 is the same as x2 doubled. So, the 2×6 has an area of 12, we double that for 4×6 to equal 24 and then we double it again for 8 x 6 for a total of 48. After doing a few of these, the students were getting pretty good at figuring out some products with their x8 math facts. I then asked them if they could figure out what 8 x 16 would be. Within a couple of minutes they had the product of 128. (Down the road I know I’ll be discussing breaking apart the 16 by the place value positions, but this flexibility of thought is exactly what I’m trying to develop.) It was awesome! The students were so proud of themselves. There is no way anyone can memorize all the possible products, but by giving them a strategy for multiplying anything times 8, they were confident they could attempt it. Loved it!