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This is a good question. There probably is not one place you will see every generalization, although some of the books I’ve written as references for teachers talk about some of them.

Examples of generalizations could be the following:

• When you subtract a number from another one, you could subtract part of it and then the rest of it if that is more convenient.
• When you add two odds, you get an even.
  When you divide by 3, the remainders rotate among 0, 1, and 2 as you go through consecutive whole numbers starting at 0.
• When you add two consecutive whole numbers, the result is always odd.
• To show the same amount of money with more coins, you simply trade a coin with a larger value for ones with smaller values, such as a dime for nickels or a quarter for nickels.
• When ax + b = cx + d, if a > c and b > d, then the solution has to be negative.

A lot of ideas for generalizations come from noticing things yourself as you and your students are working on questions. For example, if students solve 332 + 598 to get 930, you might move students toward the generalization that when you add 2 three-digit numbers, you can get a three-digit answer, but the answer is not always a three-digit number.

One of the most straightforward ways to explain this is to define a prime number as a number where exactly two multiplication equations involving whole numbers can be written with that number as their product.

For example, 5 is a prime number since the only two multiplication equations that have 5 as a product are 5 × 1 = 5 and 1 × 5 = 5.

With 1, however, there is only one equation: 1 × 1 = 1.

Another way to explain it is to define a prime number as one where there are only two possible rectangles that can be formed with that number of tiles.
For example, this method works for the number 7:

The only downside with this is that some students might argue that the rectangles are the same and conclude that there is only one rectangle. Then, when they show 1 as one square, they would also see only one rectangle.