What Do You Notice?
I was recently at a workshop where the leader brought up the base-ten area pieces on the document camera. She asked the teachers seated in the audience, “What do you notice?”
Area models and the place value number system
Once you notice one thing, notice something else. Keep going. What’s the pattern? What will the next-larger one look like? What will the next-smaller one look like? Take a moment or two—just sit with the pattern.
Maybe you notice that moving left to right, each rectangle is 1/10th the size of the previous rectangle. Maybe you notice that from right to left, each rectangle is 10 times the size of the previous one (CCSS 4.NBT.A). What relationship among operations does this demonstrate? I am blown away by the things I see when I slow down and take more time to notice. I love this subtle demonstration of inverse operations.
You may also notice the geometric shapes as they change from right to left (or vice versa), alternating between square and non-square rectangles. Why is this? Does this always happen… in all bases? Why does it always happen?
Though this series of arrays models the base-ten place value system and captures the concept of unitizing, it is conceptual in nature. The actual place value system is abstract and is difficult to model conceptually. How, after all, are we able to use only 10 numeric symbols to represent any number? I may be going out on a limb here, but I made an interesting connection after going through the fourth grade Number Corner September Calendar Grid, which takes students on a mysterious journey through an ancient Egyptian number system that does not involve place value. In this Egyptian number system, the combination of base and factor was not employed. In other words, you have an additive system that does not group “like” place values. It struck me that this is how base-ten pieces work.
In the ancient Egyptian number system 187 is one scroll, eight heel bones, and seven staffs. These symbols represent (1 x 100) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 10) + (1 x 1) + (1 x 1) + (1 x 1) + (1 x 1) + (1 x 1) + (1 x 1) + (1 x 1). It takes 16 ancient Egyptian numerals to write it! It also takes 16 base-ten pieces. In our modern system, with place value, it takes only 3 numerals. You can see the difference clearly in expanded notion: (1 x 100) + (8 x 10) + (7 x 1). Eights and sevens don't have their own symbols in the Egyptian system or the base-ten area pieces. For more information check this out.
Perhaps this is obvious, but it is interesting to note that when the area model shows 100, it literally shows 100 ones. In the 10x10 array we actually see 100 little squares. Place value, on the other hand, hides this 100 by putting the “one” in a new place. This is why we can represent 10 or 1,000 or 100,000 with different combinations of the same two symbols. Even if you consider the 10x10 a symbol, the base-ten pieces only have "symbols" for powers of 10. It seems that the base-ten area pieces function more like the Egyptian number system than the modern system.
Looking for a challenge?
Let’s consider what this model would look like for base-six. And for the sake of simplicity we can call the unit the zero term.
What does this tell us about the shape of base n when n is even? Odd? Does this pattern continue into negative exponents?
Now for an algebraic exercise… what is the pattern for the perimeter of these rectangles? Is there a general pattern that can be expressed for all bases?
Spencer is a grade 4 Bridges teacher.