Intended Audience:  Grades K-2 math teachers and parents.

In this video, 6-year-old Autumn shows different techniques for adding two numbers. Originally the plan was to concentrate on just one technique, but working with a child who already knows multiple techniques means that you just have to “go with the flow.”

The technique we were going to show is based upon the associative property. In the first problem, 7+8, Autumn breaks 8 into 3 and 5 and groups the 7 and 3 together to get 10. The answer is then simple: 10+5 or 15. This method shows off the Associative Property in algebra because we are changing the 3’s association with 5 to an association with 7. This “re-associating” is done symbolically with the parentheses below:

7+8 = 7+(3+5) = (7+3)+5 =10+5 = 15.

For teachers, parents, and students using the Eureka Math curriculum, your students practice this technique through the use of number bonds (the bond has 8 in the “whole” circle, and 3 and 5 in the two “part” circles).

Yes, it’s first grade yet Eureka Math is already preparing your children to be successful in algebra in middle school! Of course, we are not burdening students with words like “associative property” at this stage in their learning.

This is the same technique used in the last problem: 999+64. Autumn takes 1 from 64 and associates the 1 with 999 to get 1000. The answer is then easy:

999+64 = 999+(1+63) = (999+1)+63 = 1000+63 = 1063.

The second technique shows up in the second problem: 6+7. In doing this problem, Autumn says that, since 6+8=14, then 6+7=13 . She knows that 6+7 must be one less than 6+8. Enjoy the look of surprise on my face—I was definitely not ready for that response.

The third technique is easy and was used for 27+12 and 232+232. Autumn realized that, since there was no regrouping, she could add “2 tens + 1 ten = 3 tens” and “7 ones + 2 ones = 9 ones” to get 3 tens 9 ones, or 39. (One does not have to start in the farthest right ones place as in column addition.) I had to help her with the meaning of the digits, like when she said, “2+1=3” and I encouraged her to say “2 tens + 1 ten = 3 tens.”

Mathematically, this technique uses the “Any-order property,” which just means we can arrange addends in a sum in any order with any grouping we want. (It is just repeated applications of the Commutative Property and Associative Property). Symbolically,

27+12 = (20+7)+(10+2)=(20+10)+(7+2)=30+9=39,

The final technique is a combination of the previous techniques when Autumn finds 57+58:

57+58=(50+7)+(50+8)
=(50+50)+(7+8)
=100+(7+(3+5))
=100+((7+3)+5)
=100+(10+5)
=100+15
=115

Autumn, however, doesn’t do this problem exactly like that. It’s hard to tell from the video, but the method she used to find 7+8 was to break each number into 5 plus a number and use the fact that 5+5=10:

7+8=(5+2)+(5+3)=(5+5)+(2+3) = 10+5 = 15.

You get a hint that this is her method when she says “2+3=5.”

As always, comments are welcomed!

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Supported by NSF CAREER grant DMS-0748636