Posted by: degarcia | April 2, 2009

Big Idea: The Base Ten System

Base-Ten Number System

By: Miriam Coyle

The National Council of Teachers of Mathematics states that “In prekindergarten through grade 2 all students should use multiple models to develop initial understandings of place value and the base-ten number system” (p. 78). It also expects that “In grades 3-5 all students should understand the place-value structure of the base-ten number system and be able to represent and compare whole numbers and decimals” (p 148).

So, if we as teachers are expected to teach our students about the base-ten number system, what exactly is it and what is the best way to teach it for understanding?

Numeration Systems

A numeration system is a way of representing numbers using symbols. Each number must be represented by a different symbol. Otherwise, confusion as to which number was intended would be a frequent occurrence. Particular symbols used to represent numbers are known as the digits of the system. Under ideal circumstances a numeration system will: “represent a useful set of numbers (e.g. all whole numbers, integers, or real numbers), give every number represented a unique representation (or at least a standard representation), reflect the algebraic and arithmetic structure of the numbers” (Wikipedia, the free encyclopedia-numeral system).


Place-value numeration systems have limited numbers of digits which need to be repeated in order to represent larger numbers. The base of a system is the highest number to which you can count without repeating any previous digit. The number of digits in the place-value numeration system is the same number as the base of that system and must include a place-holder. The value of a number is determined by its position. The base number is how the numbers are grouped. A base of four would group by fours. The position to the right would hold up to four of the number before it would then be grouped as numbers of up to four times four, and then groups of four times four times four, and so forth.

Base-Ten

The base-ten numeration system is the system that most of the world uses today. No one knows exactly why the number ten was chosen as the base for this system, but it is theorized that it is due to the number of fingers that humans possess. Fingers are very easily assessable as counting tools. The base-ten numeration system consists of ten digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), and groups numbers into ten. Zero serves as the placeholder. It holds a position when there is nothing to be placed in a particular position. The value of a numeral in this system is determined by multiplication. Starting from the position directly to the right of the decimal point and going left, the value is (1 x the numeral), (10 x the numeral), (100 x the numeral), and so on. Each position to the left is ten times the positional value of the previous position. Each position to the right of the decimal point is 1/10the position to the right. For example, a four in the second spot to the left of the decimal point would be (10 x 4), or in other words, would have the value of 40. A 4 in the second position to the right of the decimal point would have the value of 4/100 or 0.04. This is known as the Multiplicative Principle. The numbers of the base-ten system can be added to determine the total value of the number. This is called the Additive Principle.

(100 x 4) + (10 x 5) + (1 x 6) = (400) + (50) + (6) = 456

Teaching for Understanding

When students do not have a solid grasp or understanding of the base-ten number system, it can prevent students from developing sophisticated counting strategies. They may still count by one, instead of by groups of ten or hundreds. It can also cause problems when students begin to use more complex mathematics, such as the traditional algorithm of long division. If a student does not understand that the 2 in 27 actually represents 20, then all sorts of problems can begin to surface in more advanced mathematics, resulting in frustration for both student and teacher.

    • Teachers need to help their students see numbers as groups of tens and ones. A student’s ability to count to a large number does not show understanding of the ability to group numbers into tens and ones. A great way to teach and give students experience with grouping in this way is through the use of ten-frames. Ten frames are a way for students to keep track and visually see what makes a ten, and then see what is left.

base-ten-numeration-0033

Notice the grouping of the tens. It is obvious to students that there are two groups of ten with some left over, in this case three. Students can readily see that 23 is made up of two tens and three ones. This also proves beneficial in helping students easily add and subtract ten. By adding a ten-frame, students would easily identify that there are now 33 counters, and by removing a ten-frame they would quickly identify 13 counters.

Students have also been shown to benefit from building a 100 chart and then finding patterns within the chart. This activity “helps to strengthen skip counting, and reinforces patterns that occur in the base-ten system” (Bahr, p. 110).

Another way to help students develop a grasp of base-ten is by the frequent use of base-ten blocks. Base-ten blocks consist of a flat (one hundred), a long (ten), and a unit (one). Students can visually see the groupings of a number when using base-ten blocks.


base-ten-numeration-0012

Students can easily see the different groupings of ten that go into a number when using base-ten blocks. In this case of the number 147, students can see that there is 1 hundred (100), 4 tens (40), and 7 ones (7). They can also see that the number 147 is made by adding all of these group values together (100+40+7=147) and should practice writing numbers in this expanded notation to aid in conceptual understanding of the base-ten numeration system.

Base-ten blocks are also great tools for helping students develop understanding of carrying and borrowing and addition and subtraction. Mathematicians prefer to refer to these operations as composing (carrying) and decomposing (borrowing) because these terms are more representative of what is actually taking place. When composing takes place, it is because there are now more than ten in a given position. In order to eliminate the over abundance, the groups of ten are then traded for one of the next order of grouping, in a one to one correspondence (one group of ten= one in the next position).

An example of using base-ten blocks to visually represent addition: If Mary had 9 pieces of gum and John gave her 6 more, then Mary now has a group of ten ones and five ones. She would then trade the ten ones for one ten-block (long), moving the numbers to the next position (composing), just like putting a one over the tens spot in the traditional algorithm.

base-ten-numeration-0051

An example of using base-ten blocks to visually represent subtraction: If Mary had 15 pieces of gum (represented by one ten-block and 5 units) and gave 7 to John, she would need to take the ten and break it apart into ones in order to give enough to John. This is comparable to when with the traditional algorithm of subtraction, borrowing takes place (decomposing).

base-ten-numeration-006

Another great tool giving student exposure and practice with the base-ten system is through the use of The National Library of Virtual Manipulatives. Students can practice adding and subtracting with virtual base-ten blocks, and can develop better understanding adding and subtracting using the decimal system using the virtual base-ten blocks as well.

Most of students’ confusion and frustration with math in the upper elementary grades is grounded in a lack of proper base-ten understanding. By giving students proper exposure to the use of manipulatives that promote base-ten in the younger grades, much of the heartache that goes with lack of understanding can be eliminated. Students who do show lack of understanding in the upper elementary grades should be taken back to the use of base-ten manipulatives in order to make the connections that should have been made previously. All students benefit from their use and will wean themselves from them when they are developmentally ready.


References

Bahr, Damon L., and de Garcia, Lisa Ann. (2010). Elementary mathematics is anything but elementary. Belmont, CA: Wadsworth.

Investigations in Number, Data, and Space. (2007). Math content by strand: The base-ten number system: Place value. Retrieved April 18, 2009, from http://investigations.terc.edu/library/curric-math/placevalue_2ed.pdf

KarolYeats.Com. (2008). Place value-developing understanding of numeration. Retrieved April 18, 2009, from http://www.karolyeatts.com        /Math/Numeration%20and%20Place%20Value.pdf

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: Key Curriculum Press.

Science of Everyday Things. (2009). Numeration systems. Retrieved April 18, 2009, from http://www.scienceclarified.com/Mu-Oi/Numeration-Systems.html

Wikipedia, the Free Encyclopedia. (2009). Numeral system. Retrieved April 18, 2009, from http://en.wikipedia.org/wiki/Numeral_system

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