One technique or tool that I have been using this year with the students I educate, is the mini-lesson. Usually 10 minutes in length, they are designed to develop efficient mental math strategies. They are also designed to highlight certain strategies such as the distributive property.
Mini-lessons are generally done with the entire class. There are usually 5-8 problems and they are displayed one at a time. As each problem is revealed, students are asked to solve them with their hands (they can use pencil and paper to keep track or use a model). We want the students to examine the numbers and use them to efficiently solve problems. Each problem is related in that they are helping to promote an understanding of a mathematical idea (i.e.-commutative property, associative property, etc.).
I would like to take a look at an example of a number string that we used this year with a fifth grade class. We had some students who still were not fluent with the basic multiplication facts. Facts such as 6 x 8 were not be fluent and were not automatic. As I began to use the number strings, I noticed students were able to use what they knew and apply it to what they didn’t and many have a better recall of the basic facts.
One string we issued this year involved the following number facts (remember I would only put one problem up at a time, then students would solve and discuss):
3 x 4
3 x 8
6 x 8
12 x 4
24 x 2
48 x 1
3 x 16
As you can see, the first three problems are doubles of the previous problem. Students knew that 3 x 4 was 12 and that 3 x 8 was 24. Some did not recall 6 x 8 and had to resort to a counting-on strategy. There was not much discussion here as most had recall of these facts, so I would model next to the problem an array that depicted the problem.
The next three problems help promote the doubling/halving strategy. They all have the same answer, which helped begin a discussion about equivalence. The students I was working with did not see the doubling/halving relationship between the 6 x 8 and 12 x 4. They either knew automatically or used another strategy to solve the problems. When I put the 24 x 2 on the board to solve this is when the students began to make the connection. They could see that 24 was double 12 and 2 was half of 4 and each problem had the same product. A discussion then ensued about why 6 x 8 also had the same product. Students could see either through the numbers or the arrays that 6 was half of 12 and 8 was double 4 and that they had the same products.
I had facilitated the discussion about the arrays to help them see what was happening visually to what they were explaining to the class. As you can see, in the 12 x4 and 24 x 2 problems the 12 was doubled to 24 doubling the width of the array and the 4 was halved cutting the length of the array in half as well.
The last problem was no longer following the doubling/halving strategy but tripling and thirds which led to some good discussions.
Using these types of number strings has really helped students develop fluency and efficiency when computing.