Roman Numeral Multiplication Tables

I am always intrigued by people who are multilingual. When I am asked if I speak any other languages, I always tell people, “no, I am still working on English.” But that is really not true when we look at the language of mathematics and numbers in general. In fact, different base number systems are essentially distinct languages.  For example, we are on the base 10 number system. All computer language is on a binary or base 2 system. Ancient civilizations were on base 12 and 60.

OK, what does this have to do with our lesson this week and last? When we think in Arabic numbers such as 0,1,2,3,4,5,6,7,8, and 9, but write down Roman numerals, we are translating one language to another. Multilingual people report that they often think in one language and speak in another. When I teach word problems, I encourage students to translate the English into mathematical equations.

First we explored the multiplication table by looking at a wooden table I built for my children 15 years ago. I timed each class working collaboratively to fill in all 81 squares (9x9 table). Even the young ones, could see the patterns in multiples of each single digit number. So when filling in the 5 row and columns, they could easily count 5,10,15,20,….. 1 and 2 were also easy. Three was more challenging, 4 was skip counting multiples of two, and 6,7,8, and 9 were challenging for some. The time to complete the table depended on the strategy employed. 

First, they had to divide 81 by the number of children in the group and then determine the remainder or the leftover tiles. We used several strategies for this division. Then we distributed the tiles equally to the children and had them prepare for the challenge. It was interesting to see how some children put the tiles really close to the board and ordered them from least to greatest in two rows (very orderly); other students had the tiles randomly laying around, under the board, under their legs (no organization). We opened the discussion for best practices in preparing the tiles and how they would approach the challenge once I said, “go.”

I let some of the groups employ a random approach of looking at tiles that were right in front of them. Other groups came up with a first row, first column approach which was significantly faster. The best groups communicated very well and were less focused on the tiles in front of them and more concerned with what tile was needed. The times ranged from 3-10 minutes.

Then I showed them four multiplication tables in Roman numerals:

1. I through XII
2. I,V,X,L,C,D,M
3. IV,IX,XL,XC,CD,CM
4. CREATE YOUR OWN

In class, they worked on the I through XII table in Roman numerals only. I did give them an attached answer key in Roman numerals and Arabic numbers for all three tables, but few used them. Yes, it is OK to check their answers when they are done or if they hit a wall.

With the 2nd through 5th graders we looked at strategies for multiplying numbers like XL times CD (40 x 400). Casting away the trailing zeros, we multiplied 4 x 4 to get 16 and looked at the number of place values or zeros that we casted away. In this case, we casted away three zeros so 16,000 or XVI with a horizontal line over it. They loved multiplying CM by CM so they could simply multiply 9 x 9 to get 81 and then tack on four zeros to get 810,000 or DCCCX with a horizontal line over it.

The last multiplication table is there for them to get creative by choosing their own factors to multiply.

My goal for the Ks and 1st graders is to be able to count by multiples of single digits and think in a different language: Romans. My other objective for these grades is that they could find a cell on the grid and look at the answer key to copy down the correct answer. Following horizontal and vertical rows and columns in a matrix is very challenging but this is a critical skill we will use later. Yes, it is OK if the Ks and 1st graders copy down the answers after they try it themselves.