In a previous article I explained what basic number facts are, and their importance to every student of mathematics. Once students have securely memorized their basic number facts, they should move on to figuring out extended number facts.
Review: Basic Number Facts
First, a quick review of basic number facts. These facts start with single digit addition facts (0+0 to 9+9) and the inverse facts in subtraction (0-0 to 18-9), plus multiplication facts from 0x0 up to 10×10 or 12×12 and the inverse division facts (1÷1 to 100÷10 or to 144÷12). If we don’t expect students to memorize all these facts by the time they reach 11 or 12 years old, we leave them poorly prepared for later mathematics, and in effect we encourage them to use a calculator for every calculation, since they are not equipped to recall them from memory.
Extended Number Facts
Extended facts are those facts that can easily be deduced from basic number facts by using basic principles of place value. Extended facts are found in all four operations, and knowing them or being able to find them quickly extends the student’s computational ability dramatically, by providing the tools to handle much more computation mentally. There is no upper limit to the number of extended facts that can be understood, and they will be useful no matter how many or how few a student is able to figure out.
Extended facts are deduced by taking a basic number fact and applying to the terms involved either a multiple of ten (eg, 10, 20, 30, etc) or a power of ten (10, 100, 1000, etc or 0.1, 0.01, 0.001, etc). For example, take a basic addition fact:
- 3+4 = 7
Knowing this basic number fact can be extended to figuring such facts as these:
- 13+4 = 17
- 83+4 = 87
- 30+40 = 70
- 0.03+0.04 = 0.07
In every one of these examples, the basic fact “3+4 = 7” is utilised to work out the extended fact. Providing that the student understands place value, the extended facts should be quite easy to follow. Teacher should use suitable physical or on-screen virtual materials to illustrate these ideas so that they become obvious to students: materials such as base ten blocks, bundling sticks, ten frames, or appropriate software models for numbers and operations.
Figuring Extended Facts Reduces Computation Workload
Once the teacher accepts the value of students figuring extended facts, it becomes clear that the number of computation examples for which no written or calculator method is needed expands enormously. Why should students be expected to use a written algorithm, or to reach for a calculator, to work out any of the following facts?
- 280+70 = 350 (extended from 8+7 = 15)
- 72-5 = 67 (extended from 12-5 = 7)
- 50×6 = 300 (extended from 5×6 = 30)
- 560÷8 = 70 (extended from 56÷8 = 7)
Students should be encouraged to use the best computation method which is available, each time they are called on to figure out an answer. Sometimes that method will be a written one, sometimes a calculator is best, at other times a spreadsheet is a better option. But if a mental method can be used, it will be quicker and will free the student to think about the more difficult aspects of solving problems.