The Power of Ten

Schoolchildren in the lower grades spend a lot of time learning that our number system is a place value system based on ten and powers of ten. A daily activity in the second-grade classroom in which I student taught was counting ten small “ones blocks” and replacing them with one larger ten block. Students eventually also worked with hundreds blocks, learning that ten tens could be replaced with one hundred block.

Sometimes in middle school, when students learn about ancient and medieval civilizations in their social studies class, they may learn about older number systems that did not use place value. Instead, these older number systems assigned a value to each symbol. For example, in the Roman numeral system, the number 5 had its own symbol, V; 10 had its own symbol, X; 1 its own symbol, I; 50 its own symbol, L, and so on. To represent the number 16, a Roman would use an X followed by a V and I. In our number system, 160 looks an awful lot like 16; it is just 16 multiplied by 10, and the digits shift left one place. Yet in the Roman system, the number 160 would look completely different from 16, being represented as CLX. Learning about diverse number systems can help solidify understanding of our place value-based system.

While a base ten number system like ours has some disadvantages—it has difficulty representing the fraction one-third, for example—its real strength is in the facility with which one can multiply and divide by ten, or any power of ten.

Empowering students with the ability to easily multiply and divide numbers by ten or other powers of ten is one of the most important ways I as a tutor can help students. Some students are able to grasp this concept immediately or already have mastered it, whereas some students take a bit longer, yet once they do learn this skill, their learning goes faster and their thinking becomes more flexible. Here are some of the shortcuts I emphasize:

Screen Shot 2018-04-19 at 11.11.53 AM
Shift all digits left one place by adding a zero.
Screen Shot 2018-04-19 at 1.09.29 PM
Shift all digits right one place by moving the decimal point left one place.

Later, as facility grows, the students breeze through more difficult calculations:

Screen Shot 2018-04-19 at 11.11.41 AM
I emphasize learning to “see” factors, especially 10

Learning about place value-based number doesn’t or shouldn’t stop with the early grades. Understanding decimal numbers requires a deep understanding of place value. But, really, decimal numbers are not all that different from whole numbers, at least in this respect: each place is worth ten times the place of the numeral to the right, and one tenth the value of the numeral to the left. One whole is equal to ten tenths; one tenth equals ten hundredths; one hundredth equals ten thousandths; and so on. By the end of the sixth grade students are expected to have mastered decimal numbers.

Students are introduced to scientific notation in the eighth grade, in which numbers are literally expressed as a power of ten (not using place value), a system that makes it easy to express very large or very small numbers. For example, the number 7 hundred millionths, which would take several digits to express

Screen Shot 2018-04-19 at 11.12.20 AM

is more quickly understood using scientific notation:

Screen Shot 2018-04-19 at 11.12.26 AM

The metric system, which has been adopted by U.S. scientists but not by the general public, takes advantage of the decimal basis of our number system. One meter is equal to ten decimeters; one decimeter equals ten centimeters; one centimeter equals ten millimeters; and so on.

What is your child’s understanding of our decimal and place value-based number system? Have a conversation soon.