Because kids can be smarter than we think... 
Invented algorithms 
Example 3: Unpacking a standard algorithm 
The main idea of this lesson has been that students’ invented algorithms are often based on intuitive understandings of mathematical properties. It is valuable for teachers to “unpack” the steps of their students’ thinking to verify that all steps are valid and to understand why the apparent “tricks” work. Similarly, we can analyze the more common algorithms to explore the properties that underlie the familiar steps. As Carpenter, Franke, and Loef (2003)^{1} point out, we can also consider the properties that underlie the familiar twocolumn addition algorithm. Consider the steps involved in adding 35 + 48: 
35 +48 
35 x 48

^{1} Carpenter, T., Franke, M., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary school. Portsmouth, NH: Heinemann. 
Step 1: Align the numbers in columns: 



(3 x 10 + 4 x 10) + (5 + 8) 
[base 10 notation] 


(3 + 4) x 10 + (5 + 8) 
[distributive property] 

Step 2: Add the ones, “carry the ten” then add the tens 



13 + (7 x 10) 
[commutative property] 


(1 x 10 + 3) + (7 x 10) 
[base 10 expansion] 


(1 + 7) x 10 + 3 
[distributive property] 


8 x 10 + 3 
[addition] 


83 
[base 10 compression] 

Challenge:
You can use the same “unpacking” analysis to view the steps of the common multiplication algorithm. Outline the steps and mathematical property justifications used to solve the following problem:
