Thought for the day: Use Creative Math Prompts for professional development! Gather teachers together to notice and wonder about the prompts. Leverage your own new mathematical learning to create ideas that help your students understand math more deeply!
Concepts: multi-digit multiplication; place value; multiples; multiplication patterns; even / odd numbers
Examples of noticing and wondering
I notice that every equation begins with 11.
i notice that the second factor keeps increasing by 9. (The tens digits increases by 1 and the ones digit decreases by 1.)
I notice that the answers keep increasing by 99. (The hundreds digits keeps increasing by 1, the ones digits keeps decreasing by 1, and the tens digits stays at 0.)
I notice that the sum of the digits of the second factor always equals 10.
I notice that the sum of the digits in each product always equals 11.
I wonder why the answers always increase by 99.
I wonder if the sum of the digits will always be 11 if the pattern continues.
I wonder if the patterns will change when the second factor has more than two digits.
I wonder if I would see similar patterns if each second factor was one less (18, 27, 36, etc.)
I wonder what would happen if I multiplied 11 by 109, 208, 307, 406, etc. instead.
Multiplication by 11 is a rich and fascinating topic for children (and adults!) to explore. The pattern in this image is unusual in some ways. For instance, the sum of the digits of multiples of 11 is not always 11. In fact, 209 is the smallest multiple of 11 for which the sum of the digits is odd—and the next such odd sum is 308!
Multiplication by 11 offers great opportunities for mental math. For example:
- Add 99 by adding 100 then subtracting 1. (You could also subtract 1 before adding 100.)
- Decompose numbers and use the distributive property: 11 x 19 = (10 groups of 19) plus (1 group of 19). For example:
11 x 19 = 190 + 19
11 x 28 = 280 + 28
11 x 37 = 370 + 37
This process also helps you to understand what causes the patterns.
If students write down all positive multiples of 11, they will discover even more amazing patterns, some of which point to shortcuts for multiplying by 11.
Extend the patterns in this Creative Math Prompt to discover and justify even more beautiful new patterns and relationships. The possibilities are endless!