Using pairs of compatible numbers is a great way to do mental math. Learning about compatible numbers can begin in the early grades. Students who use 10 frames (see an earlier post for copies of those) to learn about numbers to 10 can visualize the number compatible for 10 easily. For instance, looking at the 6 card, it is clearly apparent that there are 4 “empty” spots on the card, and thus 6 and 4 are compatible for 10.
As in the last post, it is easy to work on finding compatible numbers for 100 using the 100-dot array. First, using whole rows, students make the same “to ten” connection as for small numbers, but using full rows of tens. Thus 60 (or 6 full rows of 10) can be seen to be compatible with 40 (or 4 full rows of 10). From there the 100-dot array can be use for pairs of compatible numbers: 55 and 45 become compatible, etc.
Older students can work on compatible pairs for 1000. The same principle of ten works for hundreds (600 and 400 are compatible, 630 and 370 are compatible, as are 639 and 361). It is helpful for students if they examine and discuss the pattens that appear: the one’s place digits add to 10, all other place value digits add to 9).
A great way to practice finding compatible numbers is to display on the chalkboard or overhead groups of numbers, say 10 or 12, in which there are compatible pairs. You can write the numbers so that every number has a compatible partner, or have some “distractors” in the group that have no match. Students can find pairs of compatible numbers and display these on response boards.
Once students have practiced finding compatible numbers and become comfortable with that process, the skill can be used for other mental math. Consider adding 78 + 33. If a student recognizes that 78 + 22 = 100, and 33 is 11 more than 22, then 78 + 33 = 111. Using larger numbers, when doing 880 + 250, a student can split apart the second number into 2 smaller numbers, one of which is the compatible number. So 880 + 120 + 130 = 1130.
Using the same principle, students can practice finding compatible decimal numbers (e.g., 0.6 and 0.4 are compatible) use those numbers for mental computation as well.
Students should also play with fractions that are compatible (e.g., 3/8 and 5/8 are compatible, as are 13/16 and 3/16) and do some mental computation with those at the appropriate level.
There is some definite benefits in being able to find and use pairs of numbers compatible to multiple of ten (and for fractions, compatible to a whole number). I hope you will consider spending a few minutes throughout your math week working on some mental math skills with your students.