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Thought for the day: Use Creative Math Prompts for professional development. Gather teachers together to notice and wonder about the prompts. Leverage your own mathematical learning to help your students understand math more deeply!
Concepts: multi-digit multiplication; place value; multiples; multiplication patterns; even / odd numbers
Beginning
I notice that every equation begins with 11.
I notice that the second factor keeps increasing by 9. (The tens digits increase by 1 and the ones digits decrease by 1.)
I notice that the answers increase by 99. (The hundreds digits increase by 1, the ones digits decrease by 1, and the tens digits stay 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.
Exploring
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 were one less (18, 27, 36, etc.)
I wonder what would happen if I multiplied 11 by 109, 208, 307, 406, etc. instead.
Creating
As they notice and wonder, students may create their own lists of equations like the one in this prompt. Many of their ideas will flow from things that they have wondered about. For example, they may
Create new equations in the list (including ones in which the second factor has three digits.
Create new lists using other 3-digit factors multiplied by 11.
Create a lists of equations using 12 as the first factor.
Create new lists of equations based on numbers like 101, 111, 1001, 1111, etc.
For every list that students create, they should look for patterns, make and test predictions, and compare and contrast it with other lists.
Reflecting and Extending
I notice some patterns for multiplying 2-digit numbers by 11.
I wonder how these patterns extend to multiplying 3-digit numbers by 11.
I wonder how these patterns extend to multiplying 2-digits numbers by 111.
I notice that adding 9 to the second factor always increases the product by 99, because you are adding 9 more groups of 11.
I wonder what would happen if I multiplied 11 by 109, 208, 307, 406, etc. instead.
I wonder what would happen if the first number were 101 or 1001, etc.
I notice that place value helps to explain what causes these patterns.
Notes
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 multiple of 11 that does this 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
etc.
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!
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