Positional Number Systems
What happens when students don't understand the concept of positional number systems? What happens when, for some reason, the concept of zero has not been made relevant in a students educational experiences. Surely mathematics does not make sense when this happens. If a student cannot tell the difference between 105 and 15, it will be difficult for them to multiply and solve for x, among other mathematical activities.

The number zero is critical in understanding the number system that we use today. Zero helps us write numbers with ease and acts as a placeholder when there is no value to fit in that base. It allows us to write numbers accurately and effectively. Some students, however do not understand this notion. Below, you will find some strategies to use to help students adjust their understanding of positional number system.




Using Historical Numeral Systems

By using historical number systems, students will begin to compare and contrast the value of numbers. They will experience different groupings of numbers and how they additively equal different values. By having students compare and contrast various systems, they may begin to gain a deeper meaning of how our system works. This also allows them to explore the idea of expanded notation in a masked way. This strategy can be used in both a middle school and a high school setting. To demonstrate the effectiveness of this strategy, open the following document:




Creating an activity like this will help students understand both how numbers are constructed as well as what the value of zero means. Using other numbers is a good practice and provides students with a different way to think about the positional number systems we use today. If students seem to grasp this concept well, but still show evidence of some misconceptions, the next strategy can be employed.




Expressing Numbers in Expanded Notation

By having students express quantities in expanded notation, they will explore that makes a 3, 4, 5, 6 (and so on) digit number as such. Expressing numbers in expanded notation can be done in both a middle school and high school classroom, however the delivery may be varied. So, let's take a look at each individually:


Middle School:
Base ten building blocks are a good technique to use. Click here to access a virtual version of Base 10 Building Blocks. This will allow students to visualize quantities in a fast way. By expressing values using these blocks, students are using expanded notation to reason which blocks should be used and why. Perhaps an accompanying written piece, such as the word document found below, will help identify if there are gaps in a student's logic:




High School:
Depending on the student's familiarity with place value, building blocks may be a valuable learning experience. If students have never used building blocks, this may provide them with a new way to think about numbers. However, we cannot use them in such an elementary sense. It would be best to have students figure out what each building block means for themselves. The virtual building blocks may also be a good resource to use, for many high schools will not be equipped with a tangible sets of building blocks. Once students have figured out what each of the building blocks means, execute a brief activity on representing numbers with building blocks. Then, have students begin doing the abstract version. Begin by giving them a problem like such:

(1)(1000) + (5)(100) + (3)(10) + (9)(1) =

Vary these questions, and include zeros in some of the spots. This will help illustrate how zero acts as a placeholder. Then, reverse the roles. Ask students to write a variety of values in expanded notation, so they see how our place value system works.


Switching Bases

This is a strategy that can be used at both the middle school and high school level. By learning to work with numbers in base 2, it builds on or reinforces a student's understanding of place value and in turn, strengthens it. Converting bases is a skill that must be adjusted appropriately. We will look at both ways to execute this task at the middle school and high school level.

Middle School
To present this topic to middle schoolers, there are several activities that can be employed.
One of the most effective and tangible ways to teach base two to middle school students to help them reinforce their values is by creating base two building blocks for students to manipulate. The US Department of Education as provided several activities to help address the need of understanding place value. Here is the website.

Activity 1: Building Blocks

This PDF provides a handout that has all values of base 2 on it.
Counter cards (PDF - 19Kb)

The following sheet will help students organize students in writing base two numbers:
Building numbers from powers of two (PDF - 27Kb)

In writing numbers, students can use each card once, or not at all. It is important for students to note that 2^0 = 1. It will be important to make the link between base 2 and base 10, and have students compare and contrast the differences. It should also be noted that extra zeros should not be included in expressing numbers. We do not do this in base 10 and this system follows.



Activity 2: Grouping Items
Gather up toothpicks, straws, or paper clip. Have students count the item and determine the total number, in base 10. Then, have students start grouping up items in groups of 2. If there is an item left over, have the students write a 1 in the ones column. Next, have students group items in groups of 4. If there are 2 items left over, have students write a 1 in the next placeholder. If no items remain, have them write a zero. Have student repeat this process, incrementing up by powers of two until they cannot all objects in one group.

:here is a document to help you facilitate this activity.

High School

Activity 1: Converting from Base 2 to Base 10 and from Base 10 to Base 2
In a high school, based on the level of the high school class you are working with, doing activity (in the middle school category) may be a good place to start. If the classroom already understands the grouping of base 2, have students begin by exploring what 2 raised to different powers are. Once they have completed this, have students begin to write values from base 10 to base 2, by using expanded notation. The US Department of education provides this PDF: From base 10 to base 2 (PDF - 33Kb).

Next, provide students with base 2 numbers, and have them translate them to base 10 numbers. The following PDF may be of use:
Base 2 to base 10 (PDF - 32Kb)

If students find base 2 to be trivial and need more practice, have students explore counting in different bases, such as Babylonian base 60 or base 6. Students can compare and contrast this method to our base 10 method.


Activity 2: Grouping Items
Just as middle schooler were instructed how to divide up items to represent base 2 numbers, a high school variation on this activity can be developed. Once students develop an understanding for base 2, give them a collection of items, whether it be toothpicks, paper clips, marbles, etc. Have students develop a method in grouping these items in order to determine the value in base 2.

:here is a document to help you facilitate this activity.


A variation for both the middle school and high school: Have students use these unique units to group numbers!

Here's a timer to give students time to test out this activity!


Scientific Notation

A strategy that can be taught both in the middle and high school setting. Give students decimal numbers and have them represent the numbers in scientific notation. Once students are familiar with this idea, have them switch from numbers represented in scientific notation to decimal values. This will check students understanding of how a decimal point effects how the number is represented, and demonstrates the importance of zero as a place holder.

Pre Assessments, Refinement Activities, and Post Assessment Resources


Pre Assessments:
Check my place value: This site is a good self-assessment of students understanding of place value. By providing students with a list of numbers and asking to identify the place values of each number before hand, students will document their thoughts. Then, by using this website, they can monitor their understanding without running the risk of embarrassing themselves in front of the class.

Convert a Decimal Number: A conversion tool that allows students to convert base ten numbers as well as numbers expressed in scientific notation. Using this process in conjunction with an activity may help students assess their understanding of place values.

Place Value Breakdown: A good introduction to place value. It breaks down the organizational structure of place value through a visual and interactive demonstration.

Refinement Activities:

Socratic Questioning: This site explains how to use Socratic questioning to help students rationalize place value. This method should be used if there is only a slight misconception regarding place value. It can be extended to use in more-in depth activities regarding place value
http://www.garlikov.com/Soc_Meth.html

Base 2 Activities: This site provides activities using base two to develop an understanding for place value. Variations of these activities were discussed above. The activity explanations yield more depth on how teaching base two refines understanding.

Building Blocks: This site gives you the "what, where and why" as to why virtual building blocks should be used in the classroom. It highlights how they can be used in the classroom both in an individual case and as a class. Suggestions may need to be varied to fit a high school classroom.

Unusal Units : Have students convert numbers to unusual units instead of bases. This puts a fun twist on the idea!

Time: Another "base" to think about. Can there be a "zero" time? When? Perhaps a good introduction to place value and positional number systems.


Post Assessment:
Place Value Games: Have students create their own place value games, games that accentuate the major points regarding place values. Use this source as a reference for ideas to get students started.