# Complementary Numbers, Pairs, and Multiplication

Since Kakooma was such a big hit, I gave each child 16 more puzzles (4 Basic, 6 Intermediate, and 6 Advanced) of the 9 in a square variety. See attached pdf.

This week, I introduced complementary numbers and complementary pairs. Complementary Numbers add up to a power of 10 such as 10, 100, 1000 and so on. For example, the complement of 8 is 2; the complement of 48 is 52; and the complement of 870 is 130. The children loved counting by powers of 10. I have then use their fingers so they put up one finger for 10, 2 for 100, 3 for 1000 and so on. There are several advantanges to this. First, they can see how many zeros are in the power of 10 by looking at how many fingers they have up. Also, when they learn exponents, 10 to the power of 5 is a one followed by five zeros or 100,000. We counted powers of 10 all the way up to one hundred decillion. The English names of the first 10,000 powers of 10 can be found on the site: http://www.isthe.com/chongo/tech/math/number/tenpower.html.

Complementary Pairs are numbers whos units digits (or ones digits) are Complementary Numbers but all of the other digits are the same. Examples of complementary pairs are 14 and 16, 92 and 98, 113 and 117, 2009 and 2001. I did not want to confuse them with Complementary Pairs that end in zero. For example, the complementary pair of 20 is technically 30, but let's stay away from those numbers.

The attached pdf files were given to the children to practice both complementary numbers and pairs. The pages where there are blanks are intended for the children to generate their own one, two, and three digit numbers to then find complementary numbers or pairs. In class, we used double 10-sided dice which generated these numbers. The last page of each of the pdfs have a net where they can create their own 10-sided dice. The name of this solid is called generally, a "decahedron" but the specific name is a "pentagonal dipyramid." The second to last page in each pdf is a list of randomly generated two digit numbers.

They should use the appropriate K-1st Grade or 2-6th Grade pdfs to practice both concepts.

For the advanced 2nd graders and up, introduced an algorithm where you can multiply complementary pairs in a matter of seconds. The back page of the Multiplication packet has a times table to assist when they need it. To generate the last two digits of product of two complementary numbers, simply multiply the units digits as a two digit product. For example, 1x9=09, 2x8=16, 3x7=21, 4x6=24, 5x5=25, and that is it. The first digits of your answer is found by taking the tens digit of the complementary pair and multiplying it by the next consecutive integer. For example, to multiply 98 x 92, simply multiply 9x10=90 and 8x2=16 and you have the answer 9,016. To find the first digits of product of 2-digit complementary pairs, you only need to know 9 products. They are called oblong numbers: 1x2=2, 2x3=6, 3x4=24, 4x5=20, 5x6=30, 6x7=42, 7x8=56, 8x9=72, 9x10=90.

The second graders who feel comfortable and the 3-6th graders should try doing as many as possible. My advanced students tried 3-digit complementary pairs such as 104 x 106 (multiply 10x11=110 and 4x6=24) which is 11,024. Remember, this algorithm only works for multiplying complementary pairs.

For those 1-3rd graders who are not working with the multiplication, they should add the complementary pairs together. For example, 28 + 22 = 40 + 10 = 50.

AttachmentSize
Kakooma_Level_12and_3.pdf431.6 KB
10-sided_dice.pdf28.12 KB
Complementary_Numbers_and_Pairs_K-1st.pdf60.48 KB
Complementary_Numbers_and_Pairs_2-6th.pdf88.26 KB
Complementary_Pairs_Multiplication.pdf165.5 KB