## Friday, July 10, 2009

### MKN1f - Five and ten as benchmarks

MKN1. Students will connect numerals to the quantities they represent.
f. Estimate quantities using five and ten as a benchmark. (e.g. 9 is one five and four more. It is closer to 10, which can be represented as one ten or two fives, than it is to five.)

Although this indicator includes the words "estimate," what it is talking about isn't really estimation in the sense of "about how big" a number is. Rather, it is more about looking at a number from different perspectives. Thus, 8 isn't just eight ones, but rather, it is three more than 5 and two less than 10 as well. From that perspective, this standard relates very closely to MKN2 b, "Build number combinations up to 10 (e.g., 4 and 1, 2 and 3, 3 and 2, 4 and 1 for five) and for doubles to 10 (3 and 3 for six)." Using five and ten as a benchmark is in a way a special case of this indicator. Furthermore, being able to look at numbers from multiple perspectives is something that is continuously emphasized in the elementary GPS. According to Elementary School Teaching Guide for the Japanese Course of Study, the ability to see a number as a sum, a difference, a product, or a quotient of other numbers is an important foundation for algebraic thinking.

One common tool that is often used to help students develop this idea of five and ten as a benchmark is a ten frame:

It is just a table with 2 rows of 5 cells. Different numbers can be represented by filling up these cells with counters. However, when you represent numbers 6, 7, and 8, it is important that a row is filled up completely so that those numbers are represented as 5 and some more,

not like

The latter representation is useful if we want children to develop the understanding that 8 can be represented as 4 and 4 (MKN2b). Ten frames are very versatile tools, but that means we, as teachers, must be very intentional about how we use them to help students develop a specific understanding.