81 Addition Pairs


I am working on a presentation for next week on a mathematics intervention program called Knowing Mathematics and I was thinking about the “Across-10 facts.” If students leave first grade not knowing these facts because they have not internalized them at that time do we just repeat the same procedure next year and the year after that and the year after that (more of the same facts on a worksheet, over and over again until they do internalize)?

One would hope not and this is where the differences in the approach to the teaching of mathematics will arise. As the motto of the blog, Educational Insanity states: “Insanity: doing the same thing over and over again and expecting different results.” (quote by Albert Einstein) having students do the same thing over and over again whether it is from a conceptual or procedural standpoint makes no sense. How do we provide students with the tools necessary to master what they don’t know?

So let’s assume we have a student in second grade or up who has not yet mastered the single-digit combinations of nine. Let us also assume that they have had their share or worksheets and activities that support his development. A different approach we could is to have the students look for a pattern within the numbers.


What is it that we would want the student to notice in column 1? The digit in the ones place of the sum is always one less than the second addend. Columns 1 & 2 are related facts (a.k.a-Commutative Property of Addition. So if we know one “fact” we really know two).

How is the third column is related to the first column? We would want to bring out in the students that subtraction is the inverse of addition (important to understand, not just know as students move into algebra and solve equations) and that each equation in each row is related to each other.

As you can see what goes on in the early grades supports what is happening in the later grades. Therefore we cannot rely on any one approach to teaching, but we need to use what is going to help that particular student.

By working through these patterns and applying them in different situations (i.e.-practice, problem solving, at the store. etc.) students very well may have a better understanding and mastery of these particular facts. In the end we want students to not only internalize these “across-10 facts” but to know how to apply their understandings to new situations. If we are computing 39+ 27, we know it is the same as 40+26 because we can compose 39 + 1 more from the 27 to make it 40 and 26 (all should be done mentally) which is 66. If we have 37-29, we know that we could subtract 30, which leaves us with 7 and add 1 for a difference of 8 (again done mentally).